[Bernard Maskit] Kleinian Groups (Grundlehren Der org

Grundlehren der mathematischen Wissenschaften 287

A Series of Comprehensive Studies in Mathematics

Bernard Maskit

Kleinian Groups

Springer-Verlag

Grundlehren dermathematischen Wissenschaften 287A Series of Comprehensive Studies in Mathematics

Editors

M. Artin S. S. Chern J. M. Frohlich E. HeinzH. Hironaka F. Hirzebruch L. HormanderS. MacLane C. C. Moore J. K. Moser M. NagataW. Schmidt D. S. Scott Ya. G. Sinai J. TitsM. Waldschmidt S. Watanabe

Managing Editors

M. Berger B. Eckmann S. R. S. Varadhan

Bernard Maskit

Kleinian Groups

With 67 Figures

Springer-VerlagBerlin Heidelberg New YorkLondon Paris Tokyo

Bernard Maskit

Dept. of MathematicsSUNY at Stony BrookStony Brook, NY 11794USA

Mathematics Subject Classification (1980): 30F40

ISBN 3-540-17746-9 Springer-Verlag Berlin Heidelberg New YorkISBN 0-387-17746-9 Springer-Verlag New York Berlin Heidelberg

Library of Congress Cataloging-in-Publication DataMaskit. Bernard. Kleinian groups.(Grundlehren der mathematischen Wissenschaften : 287)Bibliography: p.Includes index.I Kleinian groups. 1. Title. If. SeriesQA331.M418 1987 515 87-20632

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To Wilma

Introduction

The modern theory of Kleinian groups starts with the work of Lars Ahlfors andLipman Bers; specifically with Ahlfors' finiteness theorem, and Bers' observationthat their joint work on the Beltrami equation has deep implications for thetheory of Kleinian groups and their deformations. From the point of view ofuniformizations of Riemann surfaces, Bers' observation has the consequencethat the question of understanding the different uniformizations of a finiteRiemann surface poses a purely topological problem; it is independent of theconformal structure on the surface. The last two chapters here give a topologicaldescription of the set of all (geometrically finite) uniformizations of finiteRiemann surfaces. We carefully skirt Ahlfors' finiteness theorem. For groupswhich uniformize a finite Riemann surface; that is, groups with an invariantcomponent, bne can either start with the assumption that the group is finitelygenerated, and then use the finiteness theorem to conclude that the grouprepresents only finitely many finite Riemann surfaces, or, as we do here, one canstart with the assumption that, in the invariant component, the group representsa finite Riemann surface, and then, using essentially topological techniques,reach the same conclusion.

More recently, Bill Thurston wrought a revolution in the field by showingthat one could analyze Kleinian groups using 3-dimensional hyperbolic geome-try, and there is now an active school of research using these methods. The workhere shares some foundation with Thurston's methods, but an exploration of hisdeep and beautiful results lies beyond the scope of this book. Some of the basicmaterial developed here in Chapters II, and IV-VII is also useful as foundationfor Thurston's work.

This book was designed to be usable as a textbook for a one year advancedgraduate course in Kleinian groups. Except for Chapters III and VIII. one couldfollow the material in the order given for such a course. For the most part,Chapter III is included as a reference for some more or less well-known, relativelyeasy to derive, but hard to find facts about regular coverings of surfaces. ChapterVIII is a collection of examples of Kleinian groups with diverse properties. Theexamples range from easy illustrations in the use of combination theorems, tofairly complicated constructions of groups with esoteric properties.

For someone first learning about Kleinian groups, there are many difficultiesin the theory caused by the presence of parabolic or elliptic elements in the group.

VIII Introduction

On first reading, one should take the point of view that one is only interested inpurely loxodromic geometrically finite Kleinian groups. When read from thispoint of view, Chapters V and IX almost disappear, and Chapters VI, VII, VIII,and X become significantly shorter. After the general theory for purely lox-odromic groups is clear, then one can go back as necessary and fill in thedifficulties and complications caused by the presence of elliptic and parabolicelements.

The basic organization of the book has three levels. The highest level are thechapters; these are labelled I through X, and of course they also have names. Thenext level down are the sections, which are labelled A, B, C, ..., and these alsohave names. The lowest level consists of subsections, labelled 1, 2, 3, .... Some ofthe subsections in Chapter VIII have names, the others do not. Each subsectioncontains at most one statement of a theorem, corollary, proposition, or lemma,so internal references usually do not use the words theorem, lemma, etc. butmerely refer to the appropriate subsection. For example, from within ChapterVII, a reference to the theorem whose statement appears in Chapter VII, sectionC, subsection 2 is given as simply "C.2.". A reference to the same theorem fromoutside Chapter VII is given as "VII.C.2.".

The figures are numbered separately. These are referred to, for example, asFig. VIII.E.8; this is the 8th figure is section E of Chapter VIII; it need not haveanything to do with subsection 8 of VIII.E. Similarly, the few formulas are alsoseparately numbered.

The exercises at the end of each chapter were put in for the usual reasons, andare quite uneven in terms of difficulty (this is not to say that any of them aredeliberately unsolved or unsolvable). In broad outline, they progress accordingto the material in each chapter, but there are also some problems that were addedon at the end.

This book had its origins in 1970, when, as a Sloan Foundation Fellow at theUniversity of Warwick, I started to write a set of notes. Since then I taught acourse in Kleinian groups several times, and slowly expanded the notes until theygrew large enough to become seemingly unmanageable. Fortunately, at justabout that time, personal computers came into being, so I bought one, bought atechnical word processor, and set about rewriting and revising my manuscript. Ialso underwent some changes in my personal life, which may have something todo with my increased ability to organize myself and my notes.

During the course of the years I was writing and organizing this book, Iprofited immensely from the encouragement and advice of many friends andcolleagues. I also would never have finished this book if it were not for my wife,Wilma, whose loving patience and quiet support have been a great source ofstrength. I owe a deep debt of gratitude to my teacher, Lipman Bers, who bothtaught me what mathematical research was all about, and, through all theseyears, has been a constant source of encouragement and sound advice. I also wishto thank my long-standing friend and colleague, Irwin Kra, who always has akind word, both for his strong encouragement, and for his help in reading some

Introduction IX

of the preliminary drafts. I had help from many people, who pointed out errorsin drafts, and helped proofread; in this connection, I especially wish to thank BillAbikoff, Jim Anderson, Ara Basmajian, Andy Haas, Blaise Heltai, Peter Matel-ski, and Perry Susskind. There are many others as well, who pointed out errors ormade suggestions; I thank them all, and I hope I have not slighted anyone byfailing to mention his or her name. Of course, there are still errors remaining,hopefully none of them serious; it seems to be a general principle that no matterhow many times one goes through the manuscript, one always finds more errors.Thanks are also due to Werner Fenchel for teaching me about half-turns; thecontents of section V.B are essentially due to him. I also had help from GilbertBaumslag with the example of a locally free group. Finally, thanks are due to theNational Science Foundation for support during the many years this book was inpreparation.

Stony Brook, July 1987 Bernard Maskit

Table of Contents

Chapter I. Fractional Linear Transformations ........................... 1

I.A. Basic Concepts ...................................................... 1

1. B. Classification of Fractional Linear Transformations.............. 4I.C. Isometric Circles ..................................................... 8

1. D. Commutators ........................................................ 11

I. E. Fractional Reflections............................................... 12

I.F. Exercises ............................................................. 13

Chapter II. Discontinuous Groups in the Plane .......................... 15

II.A. Discontinuous Groups .............................................. 15

II.B. Area, Diameter, and Convergence ................................. 16

II.C. Inequalities for Discrete Groups ................................... 18

II.D. The Limit Set ........................................................ 21

II.E. The Partition of aC ................................................... 23II.F. Riemann Surfaces ................................................... 25II.G. Fundamental Domains ............................................. 29II.H. The Ford Region .................................................... 32

11.1. Precisely Invariant Sets ............................................. 35

II.J. Isomorphisms ....................................................... 36

II. K. Exercises ............................................................. 37

II.L. Notes ................................................................. 39

Chapter III. Covering Spaces .............................................. 41

III.A. Coverings ............................................................ 41

III.B. Regular Coverings .................................................. 42III.C. Lifting Loops and Regions ......................................... 45III.D. Lifting Mappings .................................................... 46

III.E. Pairs of Regular Coverings ......................................... 48III.F. Branched Regular Coverings ....................................... 49III.G. Exercises ............................................................. 51

XII Table of Contents

Chapter IV. Groups of Isometrics ......................................... 53

IV.A. The Basic Spaces and their Groups ............................... 53

IV.B. Hyperbolic Geometry .............................................. 59

IV.C. Classification of Elements of I_" ................................... 62IV.D. Convex Sets ......................................................... 65

IV.E. Discrete Groups of Isometrics ..................................... 66

IV.F. Fundamental Polyhedrons ......................................... 68

IV.G. The Dirichlet and Ford Regions ................................... 70

IV.H. Poincare's Polyhedron Theorem ................................... 73

IV.I. Special Cases ....................................................... 78

IV.J. Exercises ............................................................ 80

IV.K. Notes ................................................................ 83

Chapter V. The Geometric Basic Groups ................................. 84

V.A. Basic Signatures .................................................... 84

V.B. Half-Turns .......................................................... 85

V.C. The Finite Groups .................................................. 87

V.D. The Euclidean Groups ............................................. 91

V.E. Applications to Non-Elementary Groups ......................... 95

V.F. Groups with Two Limit Points .................................... 99

V.G. Fuchsian Groups ................................................... 103

V.H. Isomorphisms ...................................................... 109

V.I. Exercises ............................................................ Ill

V.J. Notes ................................................................ 114

Chapter VI. Geometrically Finite Groups ................................ 115

VI.A. The Boundary at Infinity of a Fundamental Polyhedron ........ 115

VI.B. Points of Approximation .......................................... 122

VI.C. Action near the Limit Set .......................................... 124

VI.D. Essentially Compact 3-Manifolds ................................. 128

VI.E. Applications ........................................................ 131

VI.F. Exercises ............................................................ 132

VI.G. Notes ................................................................ 134

Chapter VII. Combination Theorems ..................................... 135

VII.A. Combinatorial Group Theory - I ................................. 135

VII.B. Blocks and Spanning Discs ........................................ 139

VII.C. The First Combination Theorem .................................. 149

VII.D. Combinatorial Group Theory - II ................................ 156

VII.E. The Second Combination Theorem ............................... 160

VII.F. Exercises ............................................................ 168

VII.G. Notes ................................................................ 170

Table of Contents XIII

Chapter VIII. A Trip to the Zoo .......................................... 171

VIII.A. The Circle Packing Trick ......................................... 171

VIII.B. Simultaneous Uniformization .................................... 175

VIII.C. Elliptic Cyclic Constructions ..................................... 177

VIII.D. Fuchsian Groups of the Second Kind ............................ 185

VIII.E. Loxodromic Cyclic Constructions ................................ 188

VIII.F. Strings of Beads ................................................... 200VIII.G. Miscellaneous Examples .......................................... 205VIII.H. Exercises ........................................................... 210

VIII.I. Notes ............................................................... 212

Chapter IX. B-Groups ..................................................... 214

IX.A. An Inequality ...................................................... 214

IX.B. Similarities ......................................................... 216

IX.C. Rigidity of Triangle Groups ...................................... 217

IX. D. B-Group Basics.................................................... 220

IX.E. An Isomorphism Theorem ........................................ 226

IX.F. Quasifuchsian Groups ........................................... 232

IX.G. Degenerate Groups ................................................ 236

IX.H. Groups with Accidental Parabolic Transformations ............ 243

IX.I. Exercises ........................................................... 246

IX.J. Notes ............................................................... 248

Chapter X. Function Groups .............................................. 249

X.A. The Planarity Theorem ........................................... 249X.B. Panels Defined by Simple Loops ................................. 255X.C. Structure Subgroups .............................................. 258

X.D. Signatures .......................................................... 271

X.E. Decomposition .................................................... 282X.F. Existence ........................................................... 291

X.G. Similarities and Deformations .................................... 299X.H. Schottky Groups .................................................. 311

X.I. Fuchsian Groups Revisited ....................................... 314

X.J. Exercises ........................................................... 316

X.K. Notes ............................................................... 318

Bibliography ................................................................ 319

Special Symbols ............................................................. 323

Index ........................................................................ 324

Chapter I. Fractional Linear Transformations

In this chapter we review the basic properties of fractional linear transformations.For the convenience of the reader, we start from scratch and derive the propertieswe need. The point of view here is strictly one complex dimensional; isometricsof hyperbolic spaces will be developed in Chapter IV.

I.A. Basic Concepts

A.1. We start with some notation. The extended complex plane C U {cc } isdenoted by t. Every orientation preserving conformal homeomorphism of t isa fractional linear, or Mbbius, transformation; i.e., a transformation of the form

g(z) = (az + b)/(cz + d),

where a, b, c, d are complex numbers and the determinant ad - be 0 0. Wedenote the group of all fractional linear transformations by M. Every g e M iseither the identity, or has at most two fixed points.

One usually thinks of the transformation g(z) as the matrix

Ca db).

This transformation is unchanged if we multiply all four coefficients by the samenumber t # 0, while of course the matrix is changed. From here on we regardthe matrices

(cd)

and (tc td)

as being the same. More precisely, there is an isomorphism between M andPGL(2, C), the projectivized group of non-singular 2 x 2 matrices with complexentries; an easy calculation shows that composition of maps corresponds to mul-tiplication of matrices. We always regard 2 x 2 matrices as being in PGL(2, Q.

By making judicious use of our projectivizing factor t, we can ensure that thedeterminant ad - be = 1. In this case our identification establishes an iso-

2 1. Fractional Linear Transformations

morphism between Rit and PSL(2, C), the group of matrices as above, but withdeterminant 1.

Throughout this book, unless specified otherwise, we will always write ele-ments of M as matrices with determinant 1. Of course, there are two ways to dothis; that is, as elements of M,

(-ccc

db

) and Iare equal.

A.2. We are primarily interested in conjugacy classes of subgroups of M. InSL(2, C) there is essentially only one conjugation invariant function, the trace,tr( ). The trace is not well defined in tfl, but its square is; we write tr2(g) for thesquare of the trace of g; i.e., tr2(g) = (a + d)2.

A.3. Every orientation reversing conformal homeomorphism of e is of the formg(z) = (az + b)/(cz + d), ad - be 0 0. As in the orientation preserving case, thecoefficients are homogeneous, so we can always assume that the determinant,ad - be = 1. There is no standard terminology in the literature for these trans-formations; we will call them fractional reflections (we reserve the word reflectionfor a fractional reflection with a circle of fixed points). We denote the group ofall fractional linear transformations and fractional reflections by FA. We leaveit to the reader to work out the rules for translating composition in Q intomultiplication of matrices.

A.4. We sometimes call the identity element of a group the trivial element. Everynon-trivial element of IO't has either one or two fixed points; fractional reflectionsare somewhat more complicated.

We regard lines in C as being circles in C which pass through oo.

Proposition. The fixed point set of a fractional reflection is either empty, one point,two points, or a circle in C.

Proof. We write the fractional reflection as

g(z) _ (az + b)/(cz + d),

and we first assume that g(co) = oc, or equivalently, that c = 0. Using homo-geneity, we can also assume that d = 1. We now have to solve the equationz = az + b. Separating this equation into its real and imaginary parts, we gettwo inhomogeneous linear equations in two unknowns. The solution set is eitherempty, one point, or a line. Combining these with the known fixed point at oo,we obtain a fixed point set that is either one point, two points, or a circle.

We next take up the case that c # 0; assume that c = 1. We now need to solvethe equation 1z12 + dz - az - b = 0. Setting the real part equal to zero, we get

I.A. Basic Concepts 3

a circle, not necessarily of positive radius. Setting the imaginary part equal tozero, we get one, perhaps inconsistent, linear equation in two variables; its solu-tion set is either the whole plane, a line, or empty.

A.S. All the different possibilities mentioned above do occur. The transformationsz - z, z - z + 1, z -- 2z, and z - -1/z have, respectively, a circle of fixed points,one fixed point, two fixed points, and no fixed points.

It is easy to see that any point, or pair of points (or empty set) will serve asfixed point set for a fractional reflection. Right now, we need the correspondingresult for circles.

Proposition. Let L be a circle on C. Then there is a unique fractional reflection rLwhose fixed point set is L (the transformation rL is called the reflection in L; if Lis a Euclidean circle, then rL is also sometimes called inversion in L).

Proof. If L is the Euclidean circle with center a and radius p, then

rL(z) = a + _p2(z-a)'If L is the line passing through the point a, in the direction arg(z) = 0, then

the reflection is given by

rr.(z) = e2`°(z - a) + a.

Observe that ri is in M and has a circle of fixed points; so it is the identity.The same argument shows that rL is unique. For if r' is another reflection in

L, that is, r' is a fractional reflection whose fixed point set is L, then r' o rL isorientation preserving, and has a circle of fixed points; hence r' o rL = 1. Since r'and rL are both involutions, r' = rL.

A.6. Proposition. If L is any circle in C, and g is any element of IQ, then g(L) isagain a circle.

Proof. One easily sees that if g and r are any bijections of a space X, then x is afixed point of r if and only if g(x) is a fixed point of g o r o g'. In particular, g(L)is the fixed point set of go rL o g-' ; by A.4, this can only be a circle.

A.7. Let L be a circle in it, and let z be a point not on L. Then the conjugate pointz* = rL(z)

Proposition. Let g e Q, and let z and z* be conjugate points with respect to thecircle L. Then w = g(z) and w* = g(z*) are conjugate points with respect to thecircle g(L).

Proof. Let r = g o rL o g' be the reflection whose fixed point set is the circle g(L).Then r(w) = r o g(z) = g o rL o g-1 o g(z) = g o rL(z) = g(z*) = w*.


A.B. The action of h on d: is triply transitive; i.e., given any three distinct pointszt, z2, z3 on Q' and any three other distinct points wt, w2, w3, there is an elementg in Rit with g(z.) = Of course this transformation g is unique, for if we alsohave a fractional linear transformation f, with f(zm) = wm, then f -t o g has atleast three fixed points, and so is the identity.

To prove the above statement, it suffices to consider the case that wt = 0,w2 = 1, and w3 = oo. In this case, write

g(z) = Z2-Z3Z-ZtZ2-Z1 Z-Z3

A.9. There is a bijective conformal map of the extended complex plane onto the2-sphere known as stereographic projection. We think of the 2-sphere as sittingin Euclidean 3-space with its south pole at the origin of the complex plane, andits north pole at the point (0, 0, 2). The unique line between a point in C and thenorth pole passes through the 2-sphere at exactly one point; this correspondence,which has oc paired with the north pole, defines stereographic projection (seealso IV.B.1).

I.B. Classification of Fractional Linear Transformations

B.I. A fractional linear transformation with exactly one fixed point is calledparabolic. The prototypic parabolic transformation is the translation z - z + 1.

Proposition. Every parabolic element of MI is conjugate to the translation z -+ z + 1.

Proof. Let zt be the fixed point of the parabolic element g. Let z2 be some otherpoint, and let z3 = g(z2). Let f e RA map this triple of points onto oo, 0, and 1,respectively. Then h = fog of -t has its only fixed point at oo, and maps 0 to 1.It is easy to see that a transformation of the form z --> az + 1 has no finite fixedpoint if and only if a = 1. El

B.2. If g is parabolic, then tri(g) = 4. In fact, we can choose matrices for parabolicelements so that tr(g) = 2.

If g has its fixed point at oc, then we write

9=(0 1).

Proposition. If g is parabolic with fixed point x oo, then there is a unique complexnumber p 0 0 so that

(I.1) g=1 + px -px2

p 1 - px

I.B. Classification of Fractional Linear Transformations 5

Proof. We know that there is a unique matrix with determinant 1 and trace 2representing g. Write the diagonal terms as 1 + px and I - px, and note thatthese determine p. Write

_ l+px bg

c 1-pxSince the determinant is 1, be = -p2x2. The equation for the fixed points,z(cz + 1 - px) _ (1 + px)z + b, has x as its only solution; hence c = p, andb= -px2.

We remark that (I.1) is called the normal form for a parabolic element withfixed point at x.

B.3. Every fractional linear transformation with two fixed points is conjugate toone with fixed points at 0 and oo, such a transformation necessarily has the formz -+ k2z, k E C. There are two special types of such transformations; the rotationsof the form z -+ e'ez, 0 real, e'° # 1, and the dilations of the form z -+ Az, A > 0,A 1.

A transformation conjugate to a rotation is called elliptic; a transforma-tion conjugate to a dilation is called hyperbolic. A non-elliptic transformationwith exactly two fixed points is called loxodromic; these include the hyperbolictransformations.

8.4. If the transformation g has fixed points at 0 and oo, then we can writeg(z) = k2z. The number k2 is called the multiplier of g; notice that tr2(g) =(k + k-')'. If g has fixed points at two other points, then the multiplier is welldefined only after we order the two fixed points.

If g is elliptic, then the two choices for the multiplier are k2 = eie andk2 = e-!e. When these are different, we order the fixed points by imposing therequirement that k2 = ele, 0 < B < n. When k2 = - 1, there is nothing intrinsicin the transformation g to distinguish between its fixed points. In fact, the trans-formation z -+ 1/z interchanges 0 and co, and commutes with z - -z (see D.3).

We can always distinguish between the fixed points of a loxodromic trans-formation; one of them is attracting and the other repelling. For the transforma-tion g(z) = k2z, where Ik2I > 1, we have limy,- gm(z) - oc for all z # 0, so 00 isthe attracting fixed point. Similarly, 0 is the repelling fixed point. Note that theattracting fixed point of g is the repelling fixed point of g-', and vice versa.

Our next goal is to write down normal forms for the transformations withtwo fixed points. We temporarily set aside the involution, or half-turn, wherek = ± i. For all the other transformations with two fixed points, we can distin-guish between the fixed points; call one of them x and the other Y.

We conjugate x to 0, and y to oo; then we can write g in the form:

(1.2)g(z) - X - 2 z - xg(z) - y

k z - y,

In this form k2 is again called the multiplier.


Choose a square root k of the multiplier; this is well defined up to multiplica-tion by - 1. Write k2 = k/k-', and solve (1.2) for g(z) to obtain the normal form

(1 3) g =. 1,x-y k -k xk-yk-'if x and y are both oc. If x = oo, then

(I.3a)(k-t y(k - k-')),

0 k

and if y = oo, then

_ k x(k-' - k)(1.3b) g 0 k-'

If k2 = -1, and x and y are both 0 oo, choose k = i, to obtain the normalform in this case:

4) g =(I . x-y -2i i(x+y)which, up to multiplication by - 1, is symmetric in x and y. If one of the fixedpoints is at oo, call the other one x; in this case the normal form is

I.4a(i -2ix

13.5. Both the type and trace are conjugation invariant, so one expects that thereis a relation between them. Easy computations involving the normal forms:z - z + 1 and z - k2z, yield the following.

Proposition. (i) Tr2(g) is real, with 0 5 tr2(g) < 4, if and only if g is elliptic;(ii) W(g) = 4 if and only if g is either parabolic or the identity;(iii) tr2(g) is real and >4 if and only if g is hyperbolic;(iv) tr2(g) is not in the interval [0, oo) if and only if g is loxodromic, but not

hyperbolic.

B.6. For every g e M, g and g" have the same number of fixed points, unless g"is the identity. If g: 1, and g" = 1, then g is necessarily elliptic with multiplierof the form e'". In general, g and g" have the same number of fixed points,and are of the same type.

If g has two fixed points and multiplier k2, then g" has the same two fixedpoints and multiplier ken. Similarly, if g is parabolic and its normal form (I.1) hasthe parameter p, then g" is parabolic with the same fixed point, and parameter np.

B.7. It is easy to describe the action of the translation z - z + 1. Every lineparallel to the real axis is invariant under this action. The family of lines parallelto the imaginary axis is kept fixed as a family, but the translation permutes themembers of this family.

I (xk-' - yk xy(k - k-1)

1 i(x + y) 2ixyI

)

I.B. Classification of Fractional Linear Transformations 7

The general parabolic transformation with fixed point x has the same twofamilies. First, there is the family of horocycles; this is a family of tangent circlesat x; each member of this family is kept invariant by the action. Second, there isthe family of orthogonal trajectories to the first family; this is also a family oftangent circles at x. This second family of circles is kept fixed as a family, buteach circle in this family is moved along by the transformation into another suchcircle.

B.B. The action of the dilation z -+ k2z, k2 > 1, can also be described in terms ofa pair of orthogonal families of circles. The first family is the family of linesthrough the origin; not only is each such line kept invariant by the transforma-tion, but each ray of each line is kept invariant. The orthogonal trajectories tothis family is the family of Euclidean circles centered at the origin. Each of thesecircles is mapped by the transformation into another circle in the same family.

For the general hyperbolic transformation g with fixed points x and y, theelement of M, which maps 0 to x and oo to y, transforms these two families ofcircles into two families of circles related to x and Y. First, there is the family ofcircles passing through x and y. Again, not only is each such circle kept invariantby g, but each arc between x and y of each of these circles is also kept invariant.The second family of circles is the family of orthogonal trajectories to the firstfamily. We can describe this second family E by noticing that for each circleC e E, the reflection in C interchanges x and y; and if C is any circle which hasx and y as conjugate points, then C e E. That is, E is the family of circles forwhich x and y are conjugate points. The hyperbolic transformation g moves eachcircle in E to another such circle.

8.9. The Euclidean rotation centered at 0 is of the form z - e" z. The same twofamilies of circles that were used to describe the dilation can also be used todescribe the rotation, but the roles of the families are interchanged. The circlescentered at 0 are now each kept invariant, while the rays emanating from theorigin are interchanged.

For the general elliptic transformation with fixed points at x and y, the familyof circles passing through x and y is kept invariant as a family, with each circlein the family being moved along by the transformation to another circle of thisfamily. The family of orthogonal trajectories, that is, the family of circles for whichx and y are conjugate points, is elementwise fixed; each circle of this family isinvariant under the elliptic transformation.

An elliptic element of order 2 is called a half-turn. Note that if g is a half-turn,then g keeps each circle passing through its fixed points invariant. The fixedpoints cut each of these circles into two arcs that are interchanged by g.

8.10. The general loxodromic transformation with fixed points at 0 and oa hasthe form z - peiez, p > 0, p # 1. The family of circles centered at the origin is keptinvariant as a family, but no member of this family is kept invariant. The lines

8 I. Fractional Linear Transformations

through the origin are also kept invariant as a family; in general, no one line iskept invariant, but there are two exceptions. The first exception, the hyperboliccase, when eie = 1, has already been discussed. The second exception, whene'B = -1 is called demi-hyperbolic. In the demi-hyperbolic case, each line throughthe origin is kept invariant, but the two rays of the line are interchanged.

B.11. Just as lines in C are special cases of circles, so also half-planes are specialcases of discs (in this chapter, every disc is circular; in later chapters, we will usethe word "disc" to refer to a topological disc).

Proposition. The transformation g keeps some disc invariant if and only if tr2(g) >_0.

Proof. The parabolic transformation z - z + 1 keeps every half-plane Im(z) > ainvariant. The elliptic transformation with fixed points at 0 and oo keeps everydisc {IzI < a} invariant. The hyperbolic transformation with fixed points at 0and oo keeps every half-plane bounded by a line through the origin invariant.

It remains to show that a transformation g which is loxodromic but nothyperbolic has no invariant disc. We can assume that g has fixed points at 0 andco. If g has an invariant disc, then the circle on the boundary of the disc, call itL, is also invariant. Since the iterates of a point on L accumulate to the fixedpoints of g, L passes through the fixed points of g. We observed above that aEuclidean line through the origin is invariant under a loxodromic transformationg if and only if g is either hyperbolic or demi-hyperbolic. The hyperbolic case iseliminated by hypothesis. A demi-hyperbolic transformation keeps every linethrough the origin invariant, but it interchanges the two rays of each line. Sinceit preserves orientation, it keeps neither disc bounded by any of these linesinvariant.

B.12. If g is parabolic, hyperbolic, or elliptic, then we have in fact shown thatthere is a whole family of circles which bound invariant discs.

Proposition. Let g be an element of R with tr2(g) Z 0 and let z be a point whichis not fixed by g. Then there is a circle L through z, where L, and both discs boundedby L, are invariant under g.

I.C. Isometric Circles

C.I. Let

bg=(ad)

I.C. Isometric Circles 9

be some element of btl; we assume throughout this section that g(oo) 0 ao, orequivalently, that c 96 0. The point a = a(g) = g-' (co) is called the center of theisometric circle of g. Similarly, the point x = Y (g) = g(oo) is the center of theisometric circle of g-'.

The family of circles passing through a and oc is mapped by g onto the familyof circles passing through oo and Y. Hence the orthogonal trajectories to the firstfamily are mapped by g to the orthogonal trajectories to the second. That is, thefamily of Euclidean circles centered at a is mapped by g onto the family ofEuclidean circles centered at a'. There is a unique circle I = 1(g) in this first familywhich is mapped by g onto a circle of the same size;1 is called the isometric circleof g, and its image I' is the isometric circle of g'.

We could also have defined I by taking the derivative:

g'(z) = (cz + d)-2.

Then I is the set of points where Ig'(z)l = Icz + dl-2 = 1.It is immediate from the way I and I' are defined that I' = 1(g'), and that

a' = a(g°' ). I and I' have the same radius p = Icl-'; note that a = -d/c, and thata' = a/c.

C.2. Let p = p(g) denote reflection in the isometric circle I of g. Let q = q(g) bethe Euclidean reflection in the perpendicular bisector of the line segment betweena and cc;ifa=x,then q= 1. Set r=r(g)=go(gop)-'.

Note that r-'(oo) = qopog-t(oo) = qo p(a) = q(cc) = oc. Also r-'(a') _qo pog-'(a) = gopog-'(g(oo)) = gop(oo) = q(a) = a'.

Next observe that r-'(I') = qo pog-'(I') = qo p(1) = q(1) = 1'. The lastequality follows from the fact that q(I) and I' are both circles of the same radius,centered at Y.

If r preserves orientation, then it is necessarily elliptic. We write it in the form

(1.5) r(z)=k2(z-x)+a', IkI=1.If r is orientation reversing, then r(z) has the form (1.4), so we can write r(z) inthe form

(1.6) r(z) = k2(z - a') + a', Iki = 1.

In either case, r is a Euclidean motion with fixed points at a' and oo.

Proposition. Every g e M with g(oo) # oo, can be written in the form g = r o q o p,where p is reflection in the isometric circle of g, and r and q are Euclidean motions.

C3. Proposition. F.t is generated by reflections.

Proof. Since every element of RA is conjugate to one that does not fix oo, it sufficesto consider only those transformations which do not fix oc. Since p and q in theabove are reflections, it suffices to show that the transformations (1.5) and (1.6)are products of reflections. This is left to the reader as an exercise. 0


C.4. Reflections are orientation reversing; it is sometimes convenient to have aset of generators in M.

Proposition. At is generated by translations (transformations of the form z -+ z +a, a e C), rotations about the origin (transformations of the form z - e'Bz, 0 real),dilations (transformations of the form z - Az, 0 < A < 1, or A > 1), and thetransformation z -, 1/z.

Proof. If c = 0, write g(z) in the form

g(z) = Ialdlla/dl(z + b/a).

If c 0, then write

-1\Icl/2

1 ag(z) =

Ic12 C J z + d/c + c'

where we have used ad - be = 1. This last expression can be read as writing gas a product of first the translation z - z + d/c, then z - 1/z, then the rotationz -, -(IcI/c)2z, followed by the dilation z - cI-2z, followed finally by the transla-tion z-z+a/c.

CS. Proposition. Let g = r o q o p be as in C.2. Then(i) tr2(g) = 0 if and only if q = 1, and(ii) tr2(g) is real and positive if and only if r = 1.

Proof. We know that g has trace 0 if and only if g is a half-turn, which occurs ifand only if g(oo) = g-t(oo), which in turn occurs if and only if q = 1.

To prove (ii), if r is trivial then the line passing through a and a' bounds twodiscs, both of which are invariant under g. Then by B.11, tr2(g) z 0. If tr2(g) = 0,then q = 1, so g = p; this is impossible since g is orientation preserving, while preverses orientation.

Conversely, if tr2(g) > 0, then there is an invariant circle bounding an invari-ant disc through every non-fixed point. In particular, there is such a circle Lpassing through oo. L necessarily passes through a and a'. Both discs boundedby L are invariant under both p and q. Hence r preserves L, and preserves bothdiscs. Since r is a Euclidean rotation, and L passes through its fixed points,r= 1.

C.6. Since r and q are both Euclidean motions, if T is any set, the size (area,diameter, etc.) of g(T) is the same as the size of p(T). We are primarily interestedin the diameter of a set T c C. Using stereographic projection, we can regard Cas the 2-sphere in Euclidean 3-space; we denote the spherical diameter of T bydia(T). If T c C, then the Euclidean diameter, diaE(T) = sup Ix - yl, where thesupremum is taken over all pairs of points x, yeT.

I .D. Commutators 11

C.7. If p(z) is reflection in the circle I z I = p, then p(z) = p2/a = p2z/1212; so z andp(z) both lie on the same ray through the origin, with Ip(z)I = p2/Izl.

If p is reflection in 1, the isometric circle of g, then similarly, p(z) and z bothlie on the same ray emanating from a, the center of I, and Ip(z) - al = p2/Iz - al,where p is the radius of I.

Proposition. Let g e Rc be such that g(oo) : oo, and let T be a closed set which maycontain oo, but does not contain a = g-' (oo). Let b be the distance from at to T, andlet p be the radius of the isometric circle I of g. Then

dia5(g(T)) < 2p2/8,

and if oo e T, then

p2/6 5 diaE(g(T)).

Proof. Let x be the point of T which is closest to a. Then 6 = Ix - a1, and T liesoutside the circle of radius b centered at a. Thus p(T) lies inside the circle ofradius p2/b centered at a, from which the first inequality follows. The secondinequality follows from the fact that a lies in p(T).

I.D. Commutators

D.I. In this section we find all non-trivial solutions of the equation [f,g] =f o g o f -tog-' = 1. We start with the observation that if [ f, g] = 1, thenfog of -' = g, so f keeps invariant the fixed point set of g.

D.2. If g is parabolic with fixed point x, then f also has a fixed point at x. Inter-changing the roles of f and g, we see that f is also parabolic with fixed point x.

Of course, any two parabolic elements with the same fixed point commute.

D.3. The situation is slightly more complicated if g has two fixed points. Wenormalize (i.e., conjugate by an appropriate element of 111) so that g(z) = k2z;then either f also fixes 0 and oo, or else f interchanges these two points.

In the first case, there is nothing more to be said; if two transformations bothfix 0 and oo, then they commute.

In the second case, we also have that g interchanges the fixed points of f. Ifwe place one of the fixed points of f at 1, then f(z) = 1/z. Since g interchangesthe fixed points of f, g(z) = -z. These two transformations do commute; weobserve that they generate a non-cyclic group of order four.

We summarize the above information in the following.

Proposition. Two non-trivial elements f and g of RA commute if and only if either(i) they have exactly the same fixed point set, or(ii) each is elliptic of order 2, and each interchanges the fixed points of the other.


D.4. In addition to knowing when two elements commute, we also need sufficientconditions for the commutator of two elements to be parabolic.

Proposition. If f has exactly two fixed points and f and g share exactly one fixedpoint, then the commutator If, g] is parabolic.

Proof. Normalize the group generated by f and g so that the common fixed pointis at oo, and so that the other fixed point of f is at 0. Then write

f = (0 t-1)' g = (0 abt)'

and computea

If, g] _ (0tot) (0 ab1) (tot

t0) (a0-1

ab

(I

-ab + t2ab0 1

Since ah(t2 - I) 96 0, the transformation is parabolic.

D.5. The converse to the above is false. For example, set

=1/2 0 _ (5/3 4/3f0 2)' g 4/3 5/3)'

I.E. Fractional Reflections

E.I. Fractional reflections are classified by the number of fixed points. A trans-formation with a circle of fixed points is a reflection; a transformation withexactly two fixed points is semi-hyperbolic; a transformation with exactly onefixed point is semi-parabolic; a transformation with no fixed points is semi-elliptic.

E.2. Reflections were discussed in A.5. For our purposes here, it suffices to noticethat if g is a reflection, then g2 = 1.

E.3. If g is semi-hyperbolic, we can conjugate by an element of M so that thefixed points of g are at 0 and oo. Then g(z) = kz, for some k e C. It is easy to seethat the equation for the fixed points of g, z = kz, has a line of solutions if andonly if Iki = 1. Hence Iki # 1. Write k = pe'B, and observe that g2(z) = p2z; i.e.,q2 is hyperbolic.

E.4. If g is semi-parabolic, then conjugate g by an element of R so that its fixedpoint is at x, and so that g(0) = 1. Then g(z) = kz + 1, where k e C. Write

I.F. Exercises 13

z = x + iy, k = a + ifi, and separate the equation for the fixed points of g intoits real and imaginary parts. This yields the two equations

(1 - a)x - fly= 1,

-fix + (I +a)y=0.Since this pair of equations can have no solution, 1k!2 = 1, and k -1. Weconclude that g2(z) = Ik2Iz + I + k = z + b, b # 0; i.e., g2 is parabolic.

E.S. If g has no fixed points, then conjugate g by an element of F 1l so that g2 hasa fixed point at oo, and so that g(oo) = 0. Then g(O) = oo, so g(z) = k/z. Since ghas no fixed points, arg k # 0. Write k = peie, and observe that g2(z) = e2'8z. Weconclude that g2 is elliptic except in the case that k is real and negative, in whichcase g2 = 1.

E.6. Combining the different cases above, we have shown the following.

Proposition. An element g E M has an orientation reversing square root in FA if andonly if tr2(g) >- 0.

E.7. One can easily classify the fractional reflections up to conjugation in laA; thisexercise is left to the reader (F.13-16).

IF. Exercises

F.I. If ge kvl, and x, y, g(x), g(y) are all different from oc, then

(g(x) - g(Y))2 = g'(x)g'(Y)(x - Y)2,

where g' is the derivative of g.

F.2. Let

' g=1a' d')f=rad//

\

l\\c c'

be matrices in SL(2, Q. Then

tr(ig) + tr(fg-1) = tr(f) tr(g).

F.3. Use the result of F.1 to prove that every g e AA preserves the cross ratio ofany four distinct points of C. If all the points are finite, the cross ratio is definedby

(Z1,Z2;z3,Z4) _21 - Z4 Z3 - Z2

Z1 - Z2 Z3 - Z4

and if any one of them is infinite, the cross ratio is defined by continuity.

14 I. Fractional Linear Transformations

F.4. Every element of AA can be written as a product of at most four reflections.

F.5. An element g e RA can be written as a product of exactly two reflections ifand only if tri(g) > 0.

F.6. The isometric circles I of g, and I' of g-t, are disjoint if Itr2(g)I > 4; they aretangent if jtr2(g)I = 4; they intersect in two points if 0 < Itr2(g)l < 4; they areequal if tr2(g) = 0.

F.7. Every g e RA has a square root in M.

F.8. (a) If g is parabolic, and k 96 0 is an integer, then the equation f o g of -t = gkalways has a solution f in M.

(b) For a given k, find all such solutions f.

F.9. (a) If g e RA is loxodromic, then the equation fog of -t = gk has a solutiononly if k = ± 1.

(b) Find all solutions for k = -1.

F.10. If g is a half-turn, then for any choice of matrix g" in SL(2,C) representingg,g2 = -1.

F.11. Let { gm } be a sequence of loxodromic transformations in U with attractingfixed point xm, repelling fixed point y,,, and tr2(gm) = Tm. Assume that xm -+x,ym - y # x, and Tm - T. Then there is a g e RA so that gm(z) -+ g(z) uniformly oncompact subsets of C.

F.12. Every element g e M with tri(g) >- 0 has infinitely many orientation revers-ing square roots.

F.13. Every reflection in FA is conjugate in FA to z -+

F.14. Every semi-hyperbolic transformation is conjugate in FA to one of the formz - kz, k > 1; distinct transformations of this form are not conjugate in Q.

F.15. Every semi-parabolic transformation is conjugate in R`Fto z - i + 1.

F.16. Every semi-elliptic transformation is conjugate in ibl to one of the formz - eie/z, 0 < 0 5 ir; distinct transformations of this form are not conjugate in Q.

F.17. Every element of FA - RA(*) can be written as a product of at most threereflections.

(*) We use the old-fashioned notation A - B to denote the set theoretic difference between A and B.

Chapter II. Discontinuous Groups in the Plane

The chapter starts with a discussion of abstract discontinuous groups, but isprimarily concerned with discontinuous groups of fractional linear transforma-tions acting on the extended complex plane. We discuss some inequalities,including Jorgensen's inequality, the limit set, and fundamental domains, inparticular, the Ford region. The point of view here is primarily one complexdimensional; real higher dimensional discontinuous groups will be discussed inChapter IV.

Several of the arguments in this and subsequent chapters involve subsequences.Without further mention, we will always denote both the original sequence andthe subsequence by the same name.

II.A. Discontinuous Groups

A.1. Let X be a topological space and let G be a group of homeomorphisms ofX onto itself. We say that the action of G at a point x e X is freely discontinuous,if there is a neighborhood U of x, so that g(U) fl u = 0, for all non-trivial g e G.The neighborhood U is called a nice neighborhood of x.

Another way of saying the same thing is to say that the translates of U bydistinct elements of G (also called the G-translates of U) are disjoint.

A.2. The set of points at which the action of G is freely discontinuous is calledthe free regular set, and is denoted by °Q = °Q(G).

A.3. A subset Y c X is G-invariant, or invariant under G, if g(Y) = Y for all g e G.The set °S2 is clearly G-invariant. It is also an open subset of X; that is, if U

is a nice neighborhood, then every point of U lies in 12.

A.4. The group G divides any set Y on which it acts into equivalence classes; xand y are G-equivalent, or equivalent under G, if there is a g e G with g(x) = y. Thespace of equivalence classes is denoted by Y/G; as usual the topology on Y/G isdefined by requiring that the natural projection p: Y -+ Y/G is both continuousand open.

16 11. Discontinuous Groups in the Plane

A.5. A subgroup G c AA whose action is freely discontinuous at some point z e Cis called a Kleinian group. In this chapter we will be concerned primarily withKleinian groups, and, unless specifically stated otherwise, all groups are sub-groups of M, and all actions are actions on C.

A.6. In general, C/G is a fairly terrible space, but °Q/G is rather nice.

Proposition. °Q/G is a Hausdorff space.

Proof. Let x and y be inequivalent points of °Q. We need to find neighborhoodsU of x, and V of y, so that g(U) fl v = 0 for all g e G. We start by choosing Uand V to be disjoint nice circular neighborhoods of x and y, respectively. SinceV is nice, there is at most one translate of x in V; if necessary, we make V smallerso that no translate of x lies in V, the closure of V.

Since the G-translates of U are disjoint circular discs, the sum of theirspherical areas is less than the area of the 2-sphere, hence the spherical diameterof any sequence of them tends to zero. Since the translates of x cannot accumulateat any point of °Q (i.e., each point of °12 has a nice neighborhood which containsat most one translate of any point), only finitely many translates of U intersectV. For each g e G with g(U) fl V * 0, there is a smaller nice circular neighbor-hood U' of x with g(U') fl V = 0. After a finite number of such steps, we findthe required neighborhoods.

A.7. We are primarily interested in properties of Kleinian groups that areinvariant under conjugation in M, and it is often desirable to have some pointsconveniently located. We can choose up to three points zm, located with respectto G; also choose an equal number of convenient points, x,,, a C, and let h e Mmap each zm onto the corresponding xm. Instead of referring to the group hGh't,we will refer to this group as being the same group G, normalized so that thegiven points {zm) defined with reference to G, are at the specified points {xm}of C.

The three specified points of C are usually 0, 1, and oo. The process describedabove is called normalization.

H.B. Area, Diameter, and Convergence

B.I. Using stereographic projection, we can regard the extended complex planeas being the 2-sphere S2. We denote distance on S2 between points or sets byd(-, ), we denote the spherical diameter of a set X by dia(X), and we denote thearea of a (measurable) set X by meas(X). For our purposes, it makes no differencewhether we use the actual distance on the sphere or the chordal metric.

Except near oc, stereographic projection has bounded distortion of distances,so inside a bounded portion of the plane, the spherical metric and the Euclidean

II.B. Area, Diameter, and Convergence 17

metric are equivalent; that is, given the bounded set U, there is a constant K > 0,so that for all x, ye U,

K-'Ix - yl S d(x,y)sKIx-yl.

B.2. Let G be a Kleinian group, and let U be a nice neighborhood of some pointz e °Q. Since the translates of U are all disjoint, the sum of the (spherical) areasof the translates of U is finite. We formally state this below.

Write the generic element of a Kleinian group G as

g=c db

),

and write Y to denote summation over all elements of G; we also use ' ( )

to denote summation over all non-trivial elements of G.

B.3. Proposition. Let U be a nice neighborhood of a point z e °Q. Then

Y_ meas(g(U)) < x.

B.4. Corollary. Every Kleinian group is countable.

B.5. Theorem. If x e °Q, then

Y' ICI-4 < x.

Proof. Choose a nice neighborhood U of co of the form {zf Iz1 > p} U {oo}. Leta be the center of the isometric circle I of some non-trivial element g e G; theradius of I is Ic1 '. Since U is nice, a = g-'(x) 0 U; in fact, we can assume thatb, the Euclidean distance from a to U, is positive. Note that b < p.

From I.C.7, we have

(11.1) diaE(g(U)) ? ICI-26-1.

Since g(U) is a circular disc contained in the complement of U, there is aconstant K > 0, so that

(11.2) meas(g(U)) K-' dia' g(U).

Combine B.3 with (II.1) and (11.2) to obtainICI-4S bz diaE(g(U)) < Kp2 Y' meas(g(U)) < oo. 13

B.6. Corollary. Let be a sequence of distinct elements of the Kleinian groupG, where oo a °Q, and let pm be the radius of the isometric circle of g,,,. Then p,, -+ 0.

W. Corollary. If x e °Q(G), and z is a point of °Q which is not G-equivalent tooo, then

Ig'(Z)1` < oc.


Proof. Since z E °Q, and is not G-equivalent to oc, there is some S > 0, so thatIz-g-t(oo)l>6forallgEG.Then

YLq"(z)12 Icz + dl-4 = Y lcl 4lz + d/c1-4 :5 6-4 Y ICI -4. O

II.C. Inequalities for Discrete Groups

C.1. The group M has a natural topology, defined by saying that a sequence {gm}of elements of K converges to g e NCO if there are representative matrices inSL(2, C), gm and g, respectively, so that each entry in g`,,, converges, as complexnumbers, to the corresponding entry in g. This topology is equivalent to thetopology defined by uniform convergence on compact subsets of C.

C.2. Proposition. Let G be a non-discrete subgroup of M. Then there is a sequenceof distinct elements of G converging to the identity.

Proof. Since G is not discrete, there is a sequence of elements { gm } of G convergingto some element g e M. Normalize G so that g is either z - z + 1, or of the formz - k2z. In either case, one easily sees that gm+, og.t converges to the identity.

aC.3. Proposition. Let G be a Kleinian group. Then G is a discrete subgroup of M.

Proof. Suppose not. Then there is a sequence { gm } of distinct elements of G, withgm - 1; so gm(z) -+ z for every z. Hence for every point z e C, either there areinfinitely many of the gm with gm(z) = z, or there are infinitely many translatesof z in every neighborhood of z. In either case, z # °Q. Q

C.4. The converse to the above is false. Some examples will be constructed inVI11.G.

C.S. The proposition below is sometimes known in the literature as the Shimizu-Leutbecher lemma.

Proposition. Let G be a discrete subgroup of ADO, where G contains f(z) = z + 1.Then for every

g=(a d)eG,c

either c = O, or l cl >_ 1.

Proof. Assume there is a g e G with 0 < lcl < 1. Let go = g, and inductively definegm by gm+t = gm of o gm'. Write

II.C. Inequalities for Discrete Groups 19

and compute

am+1 = I - amcm

2bm+1 = am

zcm+1 = cm

dm+l = I + amcm.

It follows that Icml = Icl2'"; hence cm -+0. An easy induction argument showsthat laml and I dmI are both bounded by

2-K Y 10,

J=o

where K = max(lal,1).We conclude that {am}, {bm}, {cm}, and {dm} are all bounded. Hence {gm} has

a convergent subsequence; in fact gm -+ f.

C.6. In general, if G is a group where A, B, C,..., are subgroups of G, and a, b,c, ..., are elements of G, then we denote the subgroup of G generated by A, B,C..... and a, b, c,... by <A, B, C, ... , a, b, c, ... ).

Proposition. If f and g are non-trivial elements of I@A, where f is loxodromic and fand g have exactly one fixed point in common, then <f, g> is not discrete.

Proof. By I.D.4, it suffices to assume that g is parabolic. Normalize G = <f,g)so that the common fixed point of f and g is at oo, and so that the second fixedpoint of f is at 0. If necessary, replace f by f -1, so that oo is the attracting fixedpoint off. Write

f=(k 0)0k_1

and compute

IkI > 1, g=

-Zmf-mogo fm =

1 bk

01 l-.1. O

C.7. We are now in a position to prove the generic case of Jorgensen's inequality;that is, we assume that f is loxodromic. If f is parabolic, then Jorgensen'sinequality reduces to the Shimizu-Leutbecher lemma. The case where f is ellipticrequires more refined techniques, and will not be discussed here.

Theorem (Jorgensen [36]). Let f and g generate a discrete subgroup of M, wheref is loxodromic, f and g do not share a common fixed point, and g does not keepinvariant the fixed point set off Then

(11.3) Itr2(f) - 41 + Itr[f,g] - 212! 1.

20 II. Discontinuous Groups in the Plane

Before going on to the proof of this theorem, we remark that tr[ f, g] is welldefined by the convention that whatever matrix we choose to represent f, wechoose the inverse matrix to represent f -'.

Normalize G = <f g> so that

f=(o k-,), Ikl> 1, g=(a d).

Observe that a, b, c and d are all 00. First, if a = 0, then g(co) = 0, sog o f o g-' has a fixed point in common with f. Since G is discrete, this can occuronly if g o f o g-' and f have the same fixed points, which in turn can occur onlyif g(0) = oc; we have specifically ruled this out. Similarly, we cannot have d = 0.

Likewise b = 0, or c = 0 implies that f and g have a common fixed point,which we have also ruled out.

Set go = g, recursively define gm+t = gm of o g,', write

gm =

and observe that

(11.4)

We compute

and

Set

am bm

cm dm

am+, = amdmk - bmcmk-',

bm+I = ambm(k-' - k),

Cm+I = Cmdm(k - k-'),

dm+t = amdmk-' - bmcmk.

Itr2f-41=1(k+k-')2-41=1k-k-'12,

Itr[f,g] - 21 = 12ad - (k2 + k-2)bc - 21

= IbcI Ik - k-'12.

a=Itr2(f)-41+Itr[f,g]-21=(1 +Ibcl)Ik-k-'12,and assume a < 1.

Our first goal is to prove that am, bm, cm and dm are all $0, and that1 bmcm 1 < xm I bcl. Form = 0, there is nothing to prove. We compute

Ibm+ICm+11= Iambmcmdml Ik - k-'I2 = IbmCmlll + bmcmI Ik - k-'I2

<- Ibmcml(1 + Ibmcml)Ik - k-'12

< amlbcl(l + amlbcl)Ik - k-'12

< xmIbcI(1 + Ibcl)lk - k-'12

< am+' lbcl.

H.D. The Limit Set 21

As we saw above, we can have am+, = 0 if and only if dm+t = 0. By (11.4), thiscan occur only if k° = 1; since f is loxodromic, this cannot occur. Hence am+,and dm+, are both 0. That bm+, and cm+t are different from zero follows from(11.4), and the induction hypothesis.

Since I bmcml - 0, we can extract a subsequence { g, } of distinct elements ofG. Also

am+t =(I +bmcm)k-bmcmk-' =k+bmcm(k-k-) 4k,(II.S)

dm+1 = (I + bmcm)k-t - bmCmk = k-' + bmcm(k-' - k) - k'.

If we knew that bm and cm were bounded, we could extract a convergentsubsequence. The next lemma asserts that for m sufficiently large, we can con-jugate gm by some power of f so as to achieve this result.

C.B. Lemma. Suppose

k 0f _0

k-'), Ikl > 1, and g = (c d)

are elements of a subgroup G of M, where a, b, c, and d 0, and I be l < I k 12. Thenthere is an integer m so that

a' b'g,=fmogol-m= (c'' d, ,

where a' = a, Ib'I 5 1k12, Ic'I 5 1k12, and d' = d.

Proof. Observe that a' = a, b' = bk2m, c' = ck-2m, and d' = d. Assume first thatIcl > IkI2. Choose m so that Ikl2m 5 ICI 5 IkI2m+2. Then Ic'I =

Ick-2m1 S IkI2, andIb'I = Ibk2m1 <- IkI2m+2/Icl < IkI2. The case that Ibl > Ik12 is treated analogously.

O

C.9. We now complete the proof of Jorgensen's theorem. We first observe from(11.5) that there is a subsequence {gm} where the {am} are all distinct. For msufficiently large, Ibmcml < IkI2; let gm be the element obtained from gm using C.B.Since am = am and d; = dm, the elements gm are all distinct. Since the entries inthe matrices representing g;,, are uniformly bounded, we can choose a convergentsubsequence, contradicting the assumption that G is discrete. 0

II.D. The Limit Set

D.I. A point x is a limit point for the Kleinian group G if there is a point z e °Sl,and there is a sequence {gm} of distinct elements of G, with gm(z) - x. The set oflimit points is called the limit set, and is denoted by A = A(G).

22 tt. Discontinuous Groups in the Plane

Since every neighborhood of a point of A contains infinitely many translatesof some point, A fl °Q = 0.

D.2. Theorem. Let x be a limit point of the Kleinian group G. Then there is a limitpoint y, and there is a sequence {gm} of distinct elements of G, so that gm(z) - xuniformly on compact subsets of C - { y}.

Proof. Since x is a limit point, there is a point zo E °Q, and there is a sequence{ gm } of distinct elements of G, so that x. Normalize G so that zo = oo.Now choose a subsequence so that am = y; y is clearly a limit point.As in I.C.2, write each gm as r o q o p, where p is reflection in the isometric circleof gm, and q and r are Euclidean motions. The result now follows from thefollowing observations: the center of the isometric circle of gm tends toy; gm mapsthe outside of its isometric circle onto the inside of the isometric circle of itsinverse; the common radius of the isometric circles of gm and gp,' tends to 0; thecenter of the isometric circle of g;' tends to x. p

D.3. It follows from the above that the definition of a limit point depends onlyon the sequence of elements of the group, and not on the point in °Q; that is, if{ gm } is a sequence of distinct elements of G, and gm(zo) -+ x for some zo E °Q, thenthere is a subsequence so that gm(z) x, for all z e °Q.

D.4. We remark that D.2 gives no information about the point y, other than thefact that y is a limit point. We might have y = x. We also might have a sub-sequence with gm(y) - x, and we might have a subsequence with gm(y) w # x.In the latter case, we say that y is a point of approximation. These points will beexplored further in VI.B.

D.5. Theorem. Let {gm}' be a sequence of distinct elements of the Kleinian groupG. Then there is a subsequence {gm}, and there are limit points x and y so thatgm(z) -+ x uniformly on compact subsets of C - { y}.

Proof. Normalize G so that oo E °Q, and choose a subsequence { gm } so that gm(oo)converges to some point, call it x, and so that g;'(oo) converges to y. Now useD.2. p

D.6. Theorem. A is closed, G-invariant, and nowhere dense in C.

Proof. Let x E A(G), and let g c- G. Since x is a limit point, there is a sequence {gm }of distinct elements of G, and there is a point z e °Q, with gm(z) -+ x. Theng o gm(z) g(x); hence A is G-invariant.

By definition, there are points of °Q in any neighborhood of x e A. SinceA fl °Q = 0, A is nowhere dense.

II.E. The Partition oft 23

To show that A is closed, let {xm} be a sequence of points of A, with xm x.Using D.2, we can find a single point z in °Q, and we can find sequences {gm,,,}of distinct elements of G, so that gm.k(z) -- X. We can assume without loss ofgenerality that the xm are all distinct and different from x. Let 6m be the minimaldistance from xm to any other xj. For each m, choose k(m) so that d(gm,k(m)(Z), xm) <Sm/2. Then {gm.k(m)} is a sequence of distinct elements of G, with gm,k(m)(z) - X.

D.7. In general, a set S is perfect if every point of S is a point of accumulation ofother points of S. It is a standard fact about Euclidean space that a perfect subsetof the plane (or Euclidean space of any dimension) is not countable.

Theorem. If A contains more than two points, then it is perfect.

Proof. Assume that A contains at least three points. For any limit point x, thereis a sequence {gm} of distinct elements of G, and there is a ye A, with gm(z) -4 x,for all z 0 y. In particular, there is such a sequence, and there are two distinctlimit points x 1 and x2, not necessarily distinct from x, so that gm(x,) - x, andgm(x2) - x. For each m, either gm(x1) # x, or gm(x2) # x. Hence there is asequence of distinct limit points converging to x.

D.8. A Kleinian group whose limit set consists of at most two points is called anelementary group; the others are called non-elementary. See V.E for more infor-mation about limit sets of non-elementary Kleinian groups.

II.E. The Partition of C

E.I. In general, if the group G acts on the space X, and Y c X, then the stabilizerof Yin G, denoted as StabG(Y), or just Stab(Y) if there is no danger of confusion,is defined by

Stab6(Y) = {geGIg(Y) = Y}.

Note that Stab(Y) is always a subgroup of G.In the special case that Y= {x}, a single point, then Stab(x) is just the set of

elements of G which have a fixed point at x. Also, a set Y is G-invariant if andonly if Stab0(Y) = G.

E.2. We continue with the abstract setting. We say that G acts discontinuously atx e X if there is a neighborhood U of x, so that g(U) fl u = 0 for all but finitelymany g e G.

The set of points at which G acts discontinuously is called the set of dis-continuity, or regular set, and is denoted by Q = Q(G).


E.3. In general, we say that a set Y is precisely invariant under the subgroup Hin G, if

(i) H = StabG(Y), and(ii) g(Y) fl Y = 0 for all g E G - H.

Where there is no danger of confusion, we will simply say that Y is preciselyinvariant under H, or, even more simply, that Y is precisely invariant.

Note that x e °Q if and only if there is a neighborhood U of x which is preciselyinvariant under the identity in G.

E.4. We now leave the general setting; for the rest of this chapter we assume thatG is Kleinian.

Proposition. A point x e Q(G) if and only if(i) Stab(x) is finite, and(ii) there is a neighborhood U of x which is precisely invariant under Stab(x).

Proof. If G acts discontinuously at x, then H = Stab(x) is surely finite. Sinceg(U) fl u # 0 for only finitely many g e G, we can find a perhaps smallerneighborhood which we still call U, so that g(U) fl u # 0 only for g e H. Thenn g(U), where the intersection is taken over all elements of H, is a neighborhoodof x which is precisely invariant under H.

If Stab(x) is finite and U is precisely invariant under Stab(x), then of courseg(U) fl u # 0 only for the finitely many elements of Stab(x).

ES. A neighborhood U of a point x E Q is a nice neighborhood if U is preciselyinvariant under Stab(x). Since every such neighborhood contains a circularneighborhood with the same properties, we will usually assume that nice neigh-borhoods are circular.

E.6. Theorem. For any Kleinian group G, C is the disjoint union of A and S2.

Proof. If z is a limit point, then any neighborhood U of z contains infinitely manytranslates of some point, so there are infinitely many distinct elements g E G withg(U) fl u # 0. Hence Q fl A = 0.

Now assume that x 0 Q. Then for every neighborhood U of x, there areinfinitely many translates of U which intersect U. Hence we can rind a sequenceof distinct elements { gm } of G, and we can find a sequence of points {zm } so thatzm -,. x, and gm(zm) -+ x. By D.5, there is a subsequence with gm(z) -, w uniformlyon compact subsets of the complement of y, where w and y are both limit points.If x = y, then xeA; if x # y, then the points {zm} do not accumulate at y, sogm(zm) -' w: i.e., x = w E A.

E.7. In general, Q will have many connected components; the connected com-ponents of 0 are called the components of G. Since Q is open, G has at mostcountably many components.

II.F. Riemann Surfaces 25

E.8. Every point of 0 - °Q is the fixed point of an elliptic element of G.

Proposition. Q - °Q is a discrete subset of Q.

Proof. Let {zm} be a sequence of points of Q - °S2. For each m, there is anon-trivial element g, e Stab(zm). Since each g,,, has at most two fixed points, wecan choose a subsequence so that the are all distinct. Choose a subsequenceso that zm -+ w, and so that gm(z) -i- x uniformly on compact subsets of C - { y}.Since zm, either w = y, or w = x; in either case, wE A.

E.9. Corollary. °Q is dense in C.

E.10. Proposition. If there is a sequence {g,,,} of distinct elements of the Kleiniangroup G, and there is a point y in Q(G) with g,,,(y) -+ x, then x for allzEQ(G).

Proof. Normalize G so that oo c- °Q. Since y e Q, y lies outside almost all theisometric circles of the Then g,,,(y) lies inside the isometric circle of g;'. Sincethe radii of the isometric circles tends to zero, and g,,,(y) - x, the center of theisometric circle of g;' converges to x. Then, for any z e 0, z lies outside almostall the isometric circles of the g,,,, so g,.(z) lies inside the isometric circle of gm';hence gm(z) - x.

II.F. Riemann Surfaces

F.I. Classically, Kleinian groups were studied because of their connection withRiemann surfaces. More recently the connections with hyperbolic 3-manifoldshave been extensively studied by Thurston and others. In this section, we lay theanalytic foundation for the connection between Kleinian groups and Riemannsurfaces; the topological foundation appears in the next chapter. The founda-tion for the connection between Kleinian groups and 3-manifolds appears inchapter IV.

F.2. A Riemann surface S is a connected 1-dimensional complex manifold; thatis, S is a connected Hausdorff space, where every s E S has a neighborhood U,and an associated homeomorphism ': U -- C, where, when defined, the com-posite of one of these homeomorphisms, with the inverse of another, is holo-morphic. The homeomorphism ay is called a local coordinate at s. In order to dolocal complex analysis, we will not distinguish between U and ,/i(U); that is, wewill regard O(U) as being the neighborhood of s.

We need a word for the disjoint union of at most countably many Riemannsurfaces; such an object is called a (disconnected) Riemann surface. Of course, asingle Riemann surface is also a (disconnected) Riemann surface.


Two (disconnected) Riemann surfaces are conformally equivalent if there is abiholomorphic homeomorphism between them.

We will need the following facts about Riemann surfaces. Every Riemannsurface is orientable and triangulable; also (the uniformization theorem) everysimply connected Riemann surface is conformally equivalent to either the sphere,the (complex) plane, or the disc, {z I Izi < 11. One basic classification of Riemannsurfaces is into the three categories: elliptic, parabolic, or hyperbolic, accordingas the universal covering surface is respectively the sphere, the plane or the disc.

We will also need the following corollary of the uniformization theorem.Every Riemann surface with cyclic fundamental group is conformally equivalentto either the punctured disc, {zI0 < Izi < 1}, an annulus (i.e., a set of the form{z10 < r, < Izi < r2}), or the punctured plane, {zI0 < IzI < co}.

F.3. A marked (disconnected) Riemann surface is a (disconnected) Riemann surfaceS with a discrete set of special points picked out on it; each special point x, ismarked with a symbol a., where each a, is either an integer > 2, or the symboloo. We say that is the order, or branch number of Marked Riemann surfacesare sometimes also called 2-orbifolds. The complement of the special points onS is denoted by °S.

Each marked Riemann surface has a simply connected branched universalcovering, where if a, < oo, then a. is the branch number at x,,,; if ate, = oo, thenthe point x,,, is simply deleted from the marked surface. A marked Riemannsurface is elliptic, parabolic, or hyperbolic, according as its branched universalcovering is the sphere, the plane, or the disc (see III.F).

F.4. As opposed to real 2-dimensional manifolds, a Riemann surface by definitionhas no boundary. A Riemann surface homeomorphic to a compact 2-manifoldwithout boundary is closed. A Riemann surface homeomorphic to the interiorof a compact 2-manifold, with or without boundary, is called topologically finite(in some of the literature, these are called Riemann surfaces of finite type).

The disjoint union of at most finitely many closed Riemann surfaces is aclosed (disconnected) Riemann surface; likewise, the disjoint union of at mostfinitely many topologically finite Riemann surfaces is a topologically finite (dis-connected) Riemann surface.

F.S. One of the basic facts in complex analysis is that there is no holomorphichomeomorphism from the punctured disc onto any annulus, or onto the punc-tured plane.

In a compact 2-manifold, every connected component W of the boundary hasa regular neighborhood U, where 0, the interior of U, is homeomorphic to an an-nulus, the homeomorphism extends continuously to the boundary, aU, and mapsthe closure, U, homeomorphically onto a closed annulus (i.e., a set of the form{zI0 < r, 5 Izl 5 r2 < oc}). Thus every boundary component of a topologicallyfinite Riemann surface has a regular neighborhood which is conformally equiv-

II.F. Riemann Surfaces 27

alent to either a punctured disc or an annulus; the punctured plane can be elimi-nated by making the regular neighborhood smaller, and using Liouville's theorem.

Let b be a boundary component of the topologically finite (disconnected)Riemann surface S. If there is a neighborhood of b conformally equivalent to anannulus, then we say that b bounds a disc in the complement of S, or we say thatb corresponds to a removed disc. Similarly, if the neighborhood is conformallyequivalent to the punctured disc, then we say that b corresponds to a punctureon S, or we say that S is punctured at b.

Let S be a topologically finite (disconnected) Riemann surface. S is analyt-ically finite (or sometimes simply finite) if every boundary component of S hasa regular neighborhood conformally equivalent to the punctured disc.

F.6. Theorem. Let G be a Kleinian group. Then S = Q/G is a (disconnected)Riemann surface.

Proof. We first show that S is Hausdorff, the proof is essentially the same as thatgiven in A.6. We need to show that if x and y are inequivalent points of 0, thenthere are neighborhoods U of x, and V of y, whose projections are disjoint.Choose nice neighborhoods U', and V', of x and y, respectively. Since U' and V'are in Q, only finitely many translates of U' can intersect V'; we can choosesmaller neighborhoods U and V, so that no translate of U intersects V.

Since G has at most countably many components, so does S.There is no difficulty describing the complex structure at a point of Q. Let

U be a nice neighborhood of such a point z. Since pI U is injective, p-' is welldefined on p(U). Once we have chosen p-' (p(z)), it serves as a local coordinate.If U and U' are two such neighborhoods which overlap, then p- op, wheredefined, is just an element of G.

Let z e 0 - °Q, let U be a nice neighborhood of z, and let J = Stab(z). SinceJ is finite, every non-trivial element of J is elliptic; by I.D.4, the elements of J allhave a second fixed point z' in common. Normalize so that z = 0 and z' = oc.Then the elements of J are of the form z -+ e2X'M Z. Since J is finite, there is acommon denominator n, so that J = {z -+ e2R'mt"z, m = 1_., n}; i.e., J is cyclic.

The function f(z) = z" serves as a projection map from U to U/J; that is, twopoints z, and z2 of U are J-equivalent if and only if f(z,) = 1(22). The image ofU under f is again a disc centered at the origin, where f covers its image n times,except at the origin, which is covered once. On f(U), the inverse functionqt(s) = V' is the required local homeomorphism. Here, on the overlap ofneighborhoods which exclude the image of the origin, aG o o ' is an elementof G. 0F.7. Let G be a Kleinian group, and let S = S2/G. This particular representationmakes S into a marked (disconnected) Riemann surface. The special points arethe projections of the elliptic fixed points in Q. The order at the special pointx = p(z) is the order of Stab(z). The boundary components of L/G that arepunctures are the special points of order oo.


The Kleinian group G is analytically finite if S = Sl/G is a finite marked(disconnected) Riemann surface.

The important finiteness theorem of Ahlfors asserts that if G is finitelygenerated, then it is analytically finite. (The converse to this statement is false;see VIII.A.9.)

F.8. Proposition. Let G be a Kleinian group, let R be a connected G-invariant subsetof Q, where RIG is a finite Riemann surface, and let d be the component of Gcontaining R. Then the set of points of A - R is discrete in A.

Proof. Since R c d, and R and d are both G-invariant, there is a conformalembedding of RIG into A/G. Since RIG is already a finite Riemann surface, thedifference can only be a finite set of points. The preimage of a finite set of pointsin d is of course discrete.

F.9. Throughout this book, we will use the words arc, path, and curve almostinterchangeably; they have the following meanings. A path or curve is a continuousmap w: I -' S, where I = [a, b] is a closed finite interval, the image of I hasmeasure zero, and w is locally a homeomorphism of I onto its image. An arc isdefined almost identically, except that the interval I may be half-open, or open.The arc or path w is simple if the map w is injective. The path w is a loop ifw(a) = w(b). The loop w is simple if, except for the fact that w agrees at theendpoints of I, w is injective; a simple loop is also sometimes called a simpleclosed curve, or Jordan curve. The arc w is proper if the map w is proper.

We think of all of these objects: arcs, paths, curves, and loops, both as mapsand as point sets, obtained by identifying the map with its image. It will alwaysbe clear from the context whether we regard a path, for example, as a map or asa point set.

F.10. A dissection F of a marked Riemann surface S is a finite or countablecollection of proper arcs, paths and loops on S with the following properties.

(i) Every connected component of S - F is planar (i.e., homeomorphic to anopen subset of (C).

(ii) Every compact subset of S meets only finitely many elements (i.e., arcs,paths, and loops) of

(iii) Every element of F meets every other element of F in at most finitelymany points.

(iv) Every element of F is either a simple loop, or a path where each endpointis either a special point or a point of some other element of F, or a proper arc,where each endpoint either lies on the boundary of S, or is a special point, or isa point of some other element of F (An endpoint of a proper arc on the boundaryof S need not be well defined. However we require that it can be well defined; weleave it to the reader to supply the necessary definitions.)

If every connected component of S - F is simply connected, then the dis-section is full.

II.G. Fundamental Domains 29

II.G. Fundamental Domains

G.I. One cannot easily draw a picture of, or describe, a Kleinian group. Theclosest we can come to this, in general, is to draw a picture of DIG which somehowillustrates the action of G. The usual picture is given by a fundamental set orfundamental domain, which, roughly speaking, contains one point from eachequivalence class in S2, and which, in some sense, illustrates the topology of Q/G.There is no one concept which works well for all purposes; we define severaldifferent objects.

A fundamental domain D for the Kleinian group (or discontinuous subgroupof a) G is an open subset of Q satisfying the following.

(i) D is precisely invariant under the identity in G.(ii) For every z e 0, there is a g e G, with g(z) e b.(iii) The boundary of D consists of limit points of G, and a finite or countable

collection of curves; each curve lies, except perhaps for one or both of its end-points, in Q; the intersection of the curve with Q is called a side of D.

(iv) The sides are paired by G; that is, if s is a side of D, then there is a sides', not necessarily distinct from s, and there is a non-trivial element g E G, calleda side pairing transformation, with g(s) = s'. Also (s')' = s, and the side pairingtransformation, from s' to s, is g'.

(v) If {sm} is a sequence of sides of D, then the spherical diameter, dia(sm) - 0;the sides of D accumulate only at limit points.

(vi) Only finitely many translates of D meet any compact subset of Q.

G.2. The first condition says that D is disjoint from all its translates, or, equiv-alently, that no two points of D are G-equivalent, or equivalently, that theprojection map p is injective on D.

G.3. The second condition says that p maps D fl Sl onto DIG; an equivalentstatement is that Sl c U g(D), where the union is taken over all elements of G.

G.4. For aesthetic reasons, one would like the sides of D to be circular arcs, butfor some situations, it would be a burden to require this. For analytic purposes,one usually only requires that the 2-dimensional measure of a7D be zero.

G.S. Note that if there is a sides, and side pairing transformation g, with g(s) = s,then since the side pairing transformation from s' to s is g', g-' = g; that is,g2 = 1.

G.6. Let D = D fl Q. The identifications of the sides induce an equivalencerelation on B. An interior point is equivalent only to itself; if x and y lie on sidesof D, and there is a side pairing transformation g, with g(x) = y, then x and y areequivalent. Let D* be D factored by this equivalence relation.


Observe that x and y are equivalent points of D if and only if there is anelement g E G, with g(x) = y. Hence the projection p provides a natural map ofD* into Q/G: we call this map tp.

The endpoints of the sides that lie in 0 are called vertices. The sides of D arealso paired at the vertices. For each vertex x, and side s ending at x, there is aunique other side s, where s and s' both lie on the boundary of the same localcomponent of D near x (see K.19).

G.7. Lemma. If x is a point of D', then there are at most finitely many points of Dequivalent to x.

Proof. This is an immediate consequence of condition (vi.)

G.B. Theorem. to: D* -, Q/G is a homeomorphism.

O

Proof. Condition (ii) states that tp is surjective, and condition (i) states that tp,when restricted to D, is injective. Since D is open, tp I D is a local homeomorphism.

If x is an interior point of a side s, then there is a side pairing transformationg, and a side s', so that g(s) = s'. Let x' = g(x). If x' j4 x, let 8(8') be the minimumdistance from x(x') to x'(x), or to any vertex of D, or to any limit point of G, orto any fixed point of g. By condition (v), S and 6' are both positive. Choose somenumber p < min(S, 8')/2, so that the ball of radius p about x is a nice neighbor-hood of x. Choose points y, and y2 on s, where y, and y2 lie on different sidesof x on s, and they both lie inside the disc of radius p about x. Connect y, to y2by some path which lies, except for its endpoints, inside D, and which also liesinside the disc of radius p. This path defines a neighborhood U of x in D. Similarlychoose a path connecting 9(Y 1) to g(y2); this defines a neighborhood U' of x' inD. We know that every point of D n u, or D n U', is equivalent only to itself.Also, every point of s fl U is equivalent only to the corresponding point of s' fl U'.Let V = U U g-' (U'). Note that Visa neighborhood of x, and since V is containedin the ball of radius p, V is a nice neighborhood of x. Since no two points ofV - s are G-equivalent, no two points of V are G-equivalent. Hence tp, restrictedto the projection of U U U', is a homeomorphism.

If x is an interior point of s, and g(x) = x, then g2 = 1. In this case, let S bethe minimum distance from x to any vertex of D, or to any other fixed point ofg, or to any limit point of G. Then choose p, and choose y, and y2, and the pathconnecting them, as above, except that we choose y2 = g(y, ). This again definesa neighborhood U of x in D. The points of u fl D are equivalent only tothemselves; each point z of s fl U, except for x, is equivalent only to g(z), whichalso lies in U, and x is equivalent only to itself. Let V = U U g(U). Then V is aprecisely invariant neighborhood of x. Since s divides V into two parts, one ofwhich is precisely invariant under the identity, V is a nice neighborhood of x. Itfollows that tpI U is a homeomorphism.

II.G. Fundamental Domains 31

Now let x = x, be a vertex. Then there is a sides, which has x, as an endpoint.Let si be the paired side, and let g,, mapping s, onto s; be the correspondingside pairing transformation. Let x2 = g&,). Then there is another side s2 # s'1which has x2 as an endpoint. There is a corresponding side s2, and side pairingtransformation g2, with g2(s2) = s2. Set x3 = 82(x2), etc. By G.7, this processends after a finite number of steps; that is, there is a first integer n, so that s,, = s,.We construct a neighborhood of x by choosing points ym, as above, on sm, andym = gm(ym) on and choose a path T. from to Y. (z, goes from y;, to y,)so that the open sector shaped region Um, bounded by s;,,_,, rm, and sm lies in D;the corresponding closed region lies in a nice neighborhood of xm; and so thatthe U. are all disjoint. The projection U, of the union of the Um, is obviously aneighborhood of the projection of x in D*.

Note that gl' (D) abuts D along s,. Also g;' o g2' (D) abuts g;' (D) alongg i' (s2 ), etc. However, the union V, of U, , g ' (U2 ), , g.-` 1 o... o g ' neednot be a neighborhood of x. The different sets must be disjoint, except along theirboundaries, but they need not rill out a neighborhood of x. We might have thath = g,-110 o g-1 is a non-trivial element of Stab(x), mappings, onto some arch(sl) emanating from x. Since no two points of D are equivalent, either hm(V) isdisjoint from V, or It 'I V = 1, for every m. Let V' be the union of V, together withall the distinct translates of V under powers of h. Since h(V) abuts V along h(s, ),V' is a neighborhood of x. Also, since V is precisely invariant under the identityin G, Stab(x) = <h>. We conclude that V' is a nice neighborhood of x. Theobvious homeomorphism from V, modulo the identification of s, with h(s, ), ontoV'/<h> is cp I U.

We have shown that cp is surjective and a local homeomorphism; it remainsto show that it is injective. If there are two regular points x and y on OD thatare G-equivalent, then there is a g e G mapping a neighborhood of x onto aneighborhood of y. Hence there is a point of D mapped by g to a point of theneighborhood of y constructed above. Since D is precisely invariant under theidentity, g must be one of the products of side pairing transformations mentionedin the construction above. Hence cp(x) = sp(y). 0

G.9. As part of the proof above, we showed that if D is a fundamental domainfor G, and z is a special point in D, then either z is a vertex, or z is the fixed point,on a side s, of a side pairing transformation g, where g(s) = s.

G.10. Proposition. Let s be a side of a fundamental domain D ,for the Kleiniangroup G. Then p(s) is a proper arc on S = Q/G.

Proof. There is nothing to prove unless s has a limit point for an endpoint. Letx be this endpoint, let s' be the paired side, where x' is the endpoint paired withx. Since x' is unique, and the points of s are paired only with those of s', it isobvious that the projection of s is a proper arc on D*. The result now followsfrom G.8. 0


G.1 1. Corollary. Let D be a fundamental domain for the Kleinian group G, whereD has exactly one connected component for each connected component of S = Q/G.Then the projection of the sides of D forms a dissection of each component of S.

G.12. There is a technique, called cutting and pasting, for obtaining one funda-mental domain from another. Let D be a fundamental domain, which we assumehere to be connected, let x and y be two points on 8D, and let v be a path in D,with only its endpoints on the boundary, connecting x to y, where v divides Dinto two regions R and R'. Let s be the side on which x lies, (if x is a vertex,choose s to be the side on the boundary of R'), let s' be the paired side, and letg mapping s to s' be the side pairing transformation. Let D* = (D - R') U g(R').It is easy to check that D* is again a fundamental domain for G, obtained bycutting along v, and then pasting along g(s fl R'). A simple case of this procedureis illustrated in Figures VIII.C.8-9 (the domain in Fig. VIII.C.8 is cut along thedotted line; the two parts are then joined by the transformation j).

G.13. A fundamental set is a subset of °Q which contains exactly one point fromeach equivalence class of points of °Q. There is no restriction on the topologyof a fundamental set. We can complete a fundamental domain D to a fundamentalset by adding some number of points of r?D to D. A fundamental set for G whoseinterior is a fundamental domain is called a constrained fundamental set.

It is often convenient to work with fundamental sets, which of necessitycontain no elliptic fixed points, and with the complement A of °Q; A is the unionof the limit set A, and the set of elliptic fixed points in Q.

H.H. The Ford Region

H.I. Throughout this section we assume that G is a Kleinian group and that oois a point of °Q. The Ford region is only defined for Kleinian groups with thisrestriction; it can also be defined for discontinuous groups containing orientationreversing elements.

H.2. For an arbitrary element g e G, let Da be the outside of the isometric circleof g; that is,

Da= {zECJIg'(z)I < 1}U{oo},

where g' is the derivative of g.The Ford region, or Ford fundamental domain D, is defined to be the interior

of n D9, where the intersection is taken over all non-trivial elements of G.

H.3. Theorem. The Ford region D is a fundamental domain for G.

II.H. The Ford Region 33

Proof. We know that every non-trivial g e G maps the outside of its isometriccircle onto the inside of the isometric circle of g-'; hence g(D) lies inside theisometric circle of g-'. It follows that g(D) n D = 0; this is condition (i) of G. 1.

The boundary of D consists of some arcs of isometric circles, and limit points,some of which may lie on isometric circles. The sides of D are these arcs ofisometric circles.

We know from B.6 that the radii of the isometric circles tend to zero; thisyields condition (v) for a sequence of sides lying on different isometric circles; ofcourse a sequence of sides on the same isometric circle has diameter tending tozero. Since the diameter of a sequence of distinct isometric circles tends to zero,any sequence of sides can accumulate only at limit points.

H.4. Condition (vi) follows from the observation that g(D) c g(D9), which in turnis contained inside the isometric circle of g'. The radius of these isometric circlestends to zero, and since the centers are all translates of oo, they accumulate onlyat limit points.

We have also shown that if (g,,,} is a sequence of distinct elements of G, then0.

Since oo e 'Q, different elements of G have different centers of their isometriccircles; in particular, each side of D lies on the isometric circle of a unique elementof G.

H.S. Let z be a point on a side s of D, where s lies on the isometric circle of f.Since z e Q, there can be at most a finite number of other isometric circles onwhich z lies. Also, there are at most a discrete set of points in the relative interiorof s which lie on more than one isometric circle. Assume that z is a point of sthat lies on no isometric circle other than s. Then Ig'(z)I < 1 for all g differentfrom both f and the identity, and I f'(z)I = 1. Let w = f(z), and observe that foran arbitrary g e G,

(11.6) Ig'(w)I = Ig'(f(z))I = I(gof)'(z)I/If'(z)I 5 1,

and we can have equality only for g = 1, or g = f -'. We conclude that w lies inD. for all g # f ''; so w lies on the boundary of D.

For an arbitrary z on s, (11.6) still holds. Since z e Q, z lies on at most finitelymany isometric circles; then equality in (11.6) can hold for only finitely manyg e G. Hence w lies on at most finitely many isometric circles, and lies outside allothers; it follows again that w e D. We have shown that f(s) is also a side of D;this concludes the proof of (iv).

H.6. We now prove (ii). Let z be some point of Q. Then for all but finitely manyg e G, z c- Da. There is nothing to prove if z is G-equivalent to cc. Otherwise thereis an element f e G with If '(z) I maximal among all elements of G (see B.7; theproof holds for all points of 0). Let w = f(z). Exactly as in (11.6), observe thatIg'(w)I 5 1, for all g e G. Since w e 0, for any 6 > 0 there are at most finitely many


g E G with lg'(w)I z S. Hence, if Ig'(w)I <I for all nontrivial ge G, then the sameinequality holds for all points near w; we conclude that such a point is in D.

On the other hand, if there is an h e G with Ih'(w)I = I and w does not lie inD, then w must lie on the intersection of at least three isometric circles, for it isan isolated point of n ,5g.

The set of points which lie on at least three isometric circles is countable anddiscrete in Q, so p(D) misses at most countably many points of Q/G. If there areany such points, there must be one, which we assume to be p(w), lying in theclosure of p(D). Then there is a sequence of points {zm} in D, and there is asequence of distinct elements {gm} in G, with gm(zm) - w. Choose a subsequenceas in D.2. Since dia(gm(D)) -+0, w, so weal, contradicting our assump-tion that w E Q. Q

H.7. In the argument above, we looked at isolated points which are not in theclosure of D, but which are inn Dg. In this section we observe that there mayindeed be such points.

Proposition. Let the Kleinian group G he normalized so that ao e .Q. Let x be afixed point of a parabolic element of G. Then there is a point y, G-equivalent to x,where y lies outside or on all isometric circles of G.

Proof. Normalize G so that oo is still in °Q, so that x = 0, and so that f(z) _z/(z + 1) E G. Then for any

g=Ca

d I e G,

compute

* *gofog -1 = (d2

*

Let (gm) be any sequence of elements of G with distinct lower right entriesdm. Then by B.5,

E Idml-8 < cc.

Hence there is an element heG with I d I = Ih'(0)I-'I2 maximal. Using (11.6),we conclude that Ig'(h(0))I S 1 for all g e G; i.e., h(0) lies outside or on all isometriccircles of G. Of course, in this case, since h(0) is a parabolic fixed point, we knowthat h(0) actually lies on the isometric circles of all powers of h. 0

H.B. Let G be the group generated by f(z) = z/(z + 1), and g(z) = z/(iz + 1). (Thisis the familiar group generated by z - z + 1, and z - z + i, conjugated by thetransformation z - 1/z.) In this group, the origin is an isolated limit point whichlies on all isometric circles.

I1.I. Precisely Invariant Sets 35

II.I. Precisely Invariant Sets

I.I. Let G be a Kleinian group, let Y be a connected subset of Q/G, and let T lieover Y; that is, T is a connected component of p-'(Y). Then T is preciselyinvariant under Stab(T) in G. Conversely, if T is any connected subset of Q,where T is precisely invariant under its stabilizer, then T is a connected com-ponent of p-'(p(T)).

An open, connected, non-empty subset T of 0, which is precisely invariantunder its stabilizer, is called a panel, or G-panel, or, if we need to be even moreexplicit, a (J, G}-panel, where J = Stab(T).

The simplest example of a panel is a nice neighborhood of a point in 0. Alsoeach connected component of a fundamental domain is a (I, G)-panel.

1.2. There are two standard procedures for obtaining precisely invariant sets. Oneis described above; that is, start with a connected set Y in Sl/G, and let T be aconnected component of p"'(Y). Another procedure is given in the following.

Proposition. Let D be a fundamental set for the Kleinian group G, let A be somesubset of D, and let J be a subgroup of G. Set

T = U j(A).Jef

Then T is precisely invariant under J in G.

Proof. It is immediate from the definition that T is J-invariant. If there is a g e G,and a point x e T so that g(x) a T, then find elements j and j ' in J, so thatj(x) andj ' o g(x) both lie in D. Since x, j(x), and j ' o g(x) are all G-equivalent,j(x) = j' o g(x).Since j -' o j' o g fixes a point of D; it is the identity. Hence g e J.

13. An important restatement of the above is as follows.

Proposition. A subset T e °Q(G) is precisely invariant under J if and only if thereare fundamental sets Do for J and D for G, so that Do fl T = D fl T is a fundamentalset for the action of J on T

Proof. If T is precisely invariant under J, then let E be a fundamental set for theaction of J on T, and let Do be any fundamental set for J containing E. Since twopoints of T are J-equivalent if and only if they are G-equivalent, E can also beextended to a fundamental set D for G.

Next assume that there are such fundamental sets Do and D. Suppose thereis age G, and a point x e T, with g(x) e T Then there are elements j and j' e J withj(x) and f- g(x) a Do fl T = D fl T Hencej(x) = f o g(x), and j -' o j' o g = 1.

I.4. Let T be a (J, G)-panel. If g, and g2 are elements of G, then g, (T) and g2(T)are either equal or disjoint. They are equal if and only if g, and g2 lie in the sameleft coset of J; that is, there is a j e J so that g2 = g, 0i.


1.5. Proposition. Let J be a subgroup of H, which in turn is a subgroup of theKleinian group G. Suppose that B e °Q(G) is precisely invariant under J in H, andsuppose there is a fundamental set D for G, so that every point of B is H-equivalentto some point of D. Then B is precisely invariant under J in G.

Proof. Since B is precisely invariant under J in H, it surely is J-invariant. Supposethere is an element g e G, and there is a point x e B, so that g(x) a B. Then thereare elements h, h' in H, and there are points z, z' in D, so that x = h(z) andg(x) = h'(z'). This yields go h(z) = h'(z'), so z = z'. Since B C_- °Q, z E °Q, henceg o h = h'; i.e., g e H. Then since B is precisely invariant under J in H, g e J.

1.6. If Y,, .... Y are disjoint connected subsets of Q/G, and if T. is a connectedcomponent of p-' (Ym), then not only is each T. a panel, but for k # m, g(Tk) flT. = 0 for all Y E G.

In general, we say that (T, . ... . is precisely invariant under (J...... in G,if each T. is precisely invariant under Jm, and if for m : k, and for all g e G,g(Tm)fl Tk = Q.

H.J. Isomorphisms

J.1. Let cp: G - G* be an isomorphism between Kleinian groups. Since anelement of G has finite order if and only if it is elliptic, qp preserves elliptic elements.Of course, (p preserves the order of each elliptic element.

J.2. Let H be an elliptic cyclic group of finite order q. From a purely algebraicpoint of view, there are in general several possible choices of a generator for H.From a topological point of view, there are two distinguished generators, theminimal rotations. We normalize H so that it has fixed points at 0 and oc, thenevery element of H is of the form z e""P"'z. The minimal rotations are theelements z - a±2 hi z. The minimal rotations are called the geometric generatorsof H.

The geometric generators are distinguished by the fact that they maximizetr2(h) among all non-trivial elements of H.

J.3. An isomorphism (p: G - G* is called a deformation if there is a homeomor-phism f : C - C, with f o g of -' (z) = tp(g)(z) for all g e G, and for all z e C. Wealso refer to the mapping f as the deformation. More generally, if X is G-invariant, and f : X - X* is such that f o g o f -I(x) = cp(g) (x), then we say thatf induces the isomorphism gyp. For reasons having to do with Teichmiiller spacesand deformation spaces of Kleinian groups, one often requires the mapping fina deformation to be quasiconformal.

H.K. Exercises 37

J.4. Proposition. Let 47: G -+ G* be a deformation. Then g e G is parabolic if andonly if cp(g) is parabolic. Also, for every elliptic element tr2((p(g)) = tr2(g);in particular, (p maps geometric generators of elliptic cyclic subgroups onto geo-metric generators.

Proof. The first statement follows at once from the fact that parabolic elements,by definition, have exactly one fixed point.

For the second statement, let H be a maximal elliptic cyclic subgroup of G.Normalize both G and G* so that H and 4p(H) both have their fixed points at 0and oo, and so that f(1) = 1. Then H = qp(H). Also the unit circle C, and its imageunder f, are both H-invariant. Let h = e2x"9z be a geometric generator of H, andassume that gp(h)(z) = 2xipiqz. Then, in order, f(C) passes through the points 1,e2"'P/Q, e4"iP/", .... Since f(C) is a simple H-invariant loop, this is possible only ifp= ±1.

JS. In general, an isomorphism cp between Kleinian groups is called type-preserving if both cp and cp-' preserve parabolic elements, and if tr2((P(h)) = tr2(h)for every elliptic h e G.

J.6. The following is an easy exercise in the use of homeomorphisms which induceisomorphisms.

Proposition. Let G, ( be Kleinian groups. Let X e Q(G) be G-invariant, and let fbe a homeomorphism from X onto some open set in C, where f induces theisomorphism f*: G - C. Then

(i) f(X) c DA,(ii) f(X) is is-invariant; and(iii) if Y c X is precisely invariant under H c G, then f(Y) is precisely invariant

under f, (H) in 0.

H.K. Exercises

K.I. Let G be a finite group of homeomorphisms acting on a Hausdorff space X.The action of G is freely discontinuous at x e X if and only if Stab(x) = 1.

K.2. Let G be an abstract group, and let H be a subgroup. Regard G as atopological space with the discrete topology (i.e., every subset of G is open). Hacts on G by right (or left) multiplication. This action of H is freely discontinuousat every g e G.

K.3. If G is a Kleinian group with 0 and oc both in 'Q, then there is a positiveconstant K, so that for all


bg=(cd)

lal < K lbl < Kz lcl < K31d1 s K41al.

K.4. Let G be a group of real translations; that is, every element of G is of theform z - z + a, a real. Then G is Kleinian if and only if it is cyclic.

K.5. Let G be a group of rotations; that is, every element of G is of the formz -+ ei0z, 9 real. G is Kleinian if and only if it is finite and cyclic.

K.6. Let G be a subgroup of NA, where every element of G has the same two fixedpoints, call them x and y, and every element of G is loxodromic. G is Kleinian ifand only if it is cyclic.

K.7. Let x be a fixed point of a loxodromic element of a Kleinian group. Then xis a point of approximation.

K.8. If G is non-elementary, then 4(G) is uncountable.

K.9. Construct a Kleinian group G containing an elliptic element g, where onefixed point of g is in Q and the other in A.

K.10. Let G be a hyperbolic cyclic Kleinian group with fixed points at ± 1. Thenthe Ford region for G has exactly two sides.

K.1 1. Construct the Ford region for the group generated by

a) I , b) i)' and I1

1

1)' c) ( 1

2-2), d) (-1

11

/10) )

(note that the matrix in d) has determinant 0 1).

K.12. The Ford region for the group generated by

has more than two sides (again, note the determinant).

K.13. Let S be a Riemann surface, and let Fo be a finite or countable collectionof curves on S satisfying the following.

(i) Each curve in !'o is properly embedded in S.(ii) Each compact subset of S meets only finitely many elements of !'o.(iii) The union of the elements in !'o does not disconnect S.Then there is a full dissection F of S, where Fo c T.

H.L. Notes 39

K.14. Construct a Kleinian group G, with non-trivial normal subgroup H, so thatA(H) # A(G). (Compare with V.1.21).

K.15. Let H be a subgroup of finite index in the Kleinian group G. ThenA(H) = A(G).

K.16. Let cp be an isomorphism between parabolic cyclic Kleinian groups. Thencp is a deformation.

K.17. Let qp be an isomorphism between loxodromic cyclic Kleinian groups. Thencp is a deformation.

K.18. Let G and G* be cyclic Kleinian groups, where G is hyperbolic, and G* isloxodromic but not hyperbolic. Let f : C - C be a deformation of G onto G*.Show that f is not differentiable at the fixed points of G.

K.19. Construct a fundamental domain D for the group generated by the trans-formations z - z + 1, and z z + i, where the origin is a vertex at the end offour distinct sides of D.

K.20. Let D be a fundamental domain for the Kleinian group G. Suppose thereis a connected component Do of D so that every side of Do is paired with another(or the same) side of Do, and so that these side pairing transformations generateG. Then do, the component of G containing Do, is G-invariant (i.e., g(A0) = Aofor all g e G).

H.L. Notes

A.I. The usual terminology for what we call "free discontinuity" is either "dis-continuity" or "proper discontinuity", but the terminology is not standard. A.5.The precise definition of "Kleinian group" has varied with time. The definitionused here is a bit old-fashioned. In the modern definition, a "Kleinian group" isany discrete subgroup of PSL(2; C) (there is an even more modern definition, inwhich a "Kleinian group" is any discrete group of motions of hyperbolic n-space;see for example, Tukia [94]). In this modem terminology, what we here call a"Kleinian group" is called a "Kleinian group of the second kind" ("Kleiniangroup of the first kind" refers to a discrete subgroup of PSL(2, C) whose limit setis all of C).

There are many generalizations of the notion of a Kleinian group; discretegroups of isometries of hyperbolic space are discussed in Chapter IV; a discussionof the discrete groups of motions of the n-sphere, or Euclidean n-space can befound in Wolf [99]. Discontinuous groups of quasiconformal mappings have


been studied by Gehring and Martin [27], Tukia [93], [95] and others. Moregeneral topological groups of mappings have been studied by Floyd [25],Kulkarni [47], and others. C.I. A discussion of the different topologies on Mcan be found in Beardon [11 pg. 45-54]. CS. Shimizu [85]; Leutbecher [49]. Ageneralization can be found in Apanasov [9], and Wielenberg [97]. C.7. See also(I I pg. 104-108]. A generalization can be found in Brooks and Matelski (19].F.2. Basic facts about Riemann surfaces can be found in Ahlfors-Sario [8],Farkas-Kra [23], Springer [86], or any other textbook on Riemann surfaces.F.7. Ahlfors' finiteness theorem appeared in [4]; a proof can be found in Kra [43pg. 333-338].

Chapter III. Covering Spaces

This chapter is primarily a review of standard covering space theory, includingsome easy, but not so will known facts about regular coverings. There is also adiscussion of branched regular coverings, in dimension two, and a proof of thebranched universal covering surface theorem.

Throughout this chapter, all spaces are connected manifolds of some fixeddimension, which, for most of our purposes, can be taken to be two.

III.A. Coverings

A.I. In this section, we set the notation and review the basic definitions and factsabout coverings. We assume the reader is familiar with this basic material, andgive no proofs.

A map f : (7, go) - (X, xO) maps the space X into the space X, and maps thepoint x'O E A, to the point xO a X.

A.2. A covering p: (ate, go) -+ (X, xO) is a surjection where every point x e X hasa neighborhood Ux so that p-'(Ux) is a disjoint union of open sets V.,x, wherefor each a, pl Va.x is a homeomorphism onto Ux.

The space.9 is the covering space, X is the base, and p is the projection. Thepoints to and xO = p(XO) are called the base points.

If x is any point of X, the set {p-' (x) } is the fiber over x.

AA The relationship between coverings and Kleinian groups is given by thefollowing obvious fact. If G is Kleinian, and °d is a connected component of°Q, then the natural projection p: °A -- °d/StabG(°d) is a covering, in fact, aregular covering. The natural projection p: A - A/Stab(A) is a branched regularcovering; these are discussed in III.F.

A.4. Every path w on X, starting at any point x, can be lifted uniquely to a pathw on 9 (i.e., w = p o w), once we specify the starting point x in the fiber over x;

is called a 1ft of w.

42 M. Covering Spaces

If w, and w2 are homotopic paths on X, both starting at x, then the liftsw, and w2, starting at the same point z E p-I (x), are also homotopic. In particular,if w, and w2 are loops on A' at x, then they are homotopic if and only if theirprojections w, = p o w, , and w2 = p o w2 are homotopic.

An equivalent way of saying the above is that the natural induced homo-morphism p,,: n, (A', 20) - n, (X, xo) is injective.

The subgroup (9, go)) c n, (X, xo) is called the defining subgroup of thecovering.

A.5. If p: (A', go) - (X, xo), and q: (A, )20) -+ (X, xo) are coverings with definingsubgroups M and N, respectively, then there is a covering r: (', x2o) - (A,.to)with p = q o r if and only if M c N. In particular, if two coverings p: (A', go) -(X, xo) and q: (A, Jt0) -+ (X, xo) have the same defining subgroup, then there isa homeomorphism r: (A', go) - (A, J20), where p = q o r.

Two coverings with the same defining subgroup are called equivalent. We willusually not distinguish between equivalent coverings.

A.6. Given a subgroup H of n, (X, xo), there is a covering p: (X, go) -+ (X, xo)with defining subgroup H. The covering corresponding to the identity subgroupis called the universal covering; the universal covering space is simply connected.Conversely, if p: (A', go) -+ (X, xo) is a covering where the covering space A' issimply connected, then the defining subgroup for this covering is trivial, so thisis the universal covering.

A.7. The product of two loops w, and w2, on X at xo, is the loop obtained byfirst traversing w, and then w2; this product is written as w, - w2.

For any loop w on X at xo, we denote the homotopy class of w by [w].The multiplication defined by [w, ] [w2] = [w, w2] is the multiplication in thefundamental group, a, (X, xo).

III.B. Regular Coverings

B.I. A covering whose defining subgroup is normal is called a regular or normalcovering.

B.2. Proposition. Let p: (2, go) -+ (X, xo) be a regular covering and let x', be a pointin the fiber over xo. Then the coverings p: (A, go) - (X, xo) and p: (2, x`,) --+ (X, xo)are equivalent.

Proof. Let w be some loop on X at xo; then w lifts to a loop starting at R. if andonly if [w] E N. the defining subgroup of the covering based at z,,.

III.B. Regular Coverings 43

Choose a path v from go to z, . Let v = p o v' so that v is a loop on X basedat xo. Since No is normal in n, (X, xo), [w] a No if and only if [v] - [w] - [v-'] e No;that is, w lifts to a loop starting at go if and only if v - w - v-1 also lifts to a loopstarting at go. Since the lift of a loop is unique, v w v-1 lifts to a loop startingat go if and only if w lifts to a loop starting at x, . We have shown that [w] a Noif and only if [w] a N1; i.e., No = N1. 0

B.3. Corollary. Let p: (9, zo) -+ (X, xo) be a regular covering, and let w be a loopon X based at some point x. Then w lifts to a loop starting at some point in thefiber over x if and only if w lifts to a loop starting at any other point in the fiberover x.

B.4. The converse to the above is also true.

Proposition. Let p: ()7, go) - (X, xo) be a covering. If every loop based at xo, whichlifts to a loop starting at go, also lifts to a loop starting at any other point x` inthe fiber over xo, then the covering is regular.

Proof. Let [v] be some element of n 1(X, xo), and let [w] be some element of N,the defining subgroup for the covering. Let v be the lifting of v starting at go, andlet x be the endpoint of v. Observe that w lifts to a loop starting at z if and onlyif v w - v-' lifts to a loop starting at go. Since w lifts to a loop starting at z,[v.w.v-1]eN. Q

BS. If x, is some point of X, then there is an isomorphism between n1(X,xo)and n, (X, x 1); this isomorphism is canonical up to an inner automorphism.In particular, a regular covering is well defined independent of base point,either on 9 or on X. More precisely, a normal subgroup of n,(X) is welldefined independent of base point on X, and the corresponding regular coveringp: 9 - X is well defined independent of base point on 9. Where convenient, wewill drop the base points, and simply refer to a regular covering p: 2 -- X, withdefining (normal) subgroup N.

B.6. Theorem. Let p: )Z -. X be a regular covering with defining subgroup N. Thenthere is a group G of homeomorphisms of 9 onto itself, where °Q(G) = 9, and X,is G-equivalent to k2 if and only if p(z,) = p(x2). Further, if we choose basepoints, go on .9, and xo = p(go) on X, then there is a canonical isomorphism0: n1(X, xo)/N -+ G.

Proof. By B.2, for every z in the fiber over xo, there is a homeomorphismg = gX: 2 -+ 2, where g(go) = x, and p o g = p. Let G be the group generated byall these transformations g1. Since the generators of G satisfy p o g = p, this samerelation holds for every g e G. It follows at once that if x, and z2 are G-equivalent,then p(AI) = p(z2).

44 III. Covering Spaces

Now suppose there is a point 2 e X, and there are two elements g and g' ofG, with g(A) = g'(x). Let x, be any point of J'', and let w be a path from 2 to x`1.Then g(x1) is the endpoint of g(w), which in turn is the endpoint of the lifting ofw = p(w), starting at g(x). Similarly, g'(zl) is the endpoint of the lifting of w,starting at g'(x). Since g(x) = g'(x), g(21) = g'(x,). Hence an element of G withone fixed point is the identity; i.e., G acts freely on X. It also follows that everyelement of G is one of the generators gX.

Let z, and x2 be two points of $, where p(x",) = p(x2). Let w be a path fromx, to the base point xo, and let w' be the lift of w = p(w), starting at x"2. Then theendpoint x of w' is some point in the fiber over the base point xo. The elementgz e G maps go to 2, and maps w-' to the lift of w-', starting at 2. Of course,(w')-' is the lift of w-1, starting at X; hence gx(w) = iv', so gj(x,) = x2. We haveshown that two points of I are G-equivalent if and only if their projections toX are equal.

It follows at once from the definition of covering, that the action of G on 9is freely discontinuous.

We now define the homomorphism 0: n,(X,xo)-'G. Let w be a loop onX at xo, let w be its lift, starting at xo, and let x be the endpoint of w. Set0([w]) = gX. This is well defined, for if w, is homotopic to w, then their lifts arehomotopic; hence they have the same endpoint. Then 0([w,])o0([w2])(xo) =P([w,])(x2), where A. is the endpoint of the lift of wm, starting at xa. To find0([w1])(22), look at the lift w2 of w2, starting at xo, and observe that O([wl])maps go to x,, so k([wl])(22) is the endpoint of'([w1])(w2)' which is theendpoint of the lift of w2, starting at x,. We conclude that 0([w1])(22) isthe endpoint of the lift of w, w2, starting at xo; i.e., P([wl])odP([w2])(20) _0([wi ' w2])(xo). Hence 0([w, ]) o o([w2]) = 0([14'1 ] [w2])-

Of course, an element [w] a rt 1(X, x0) is in the kernel of 0 if and only if w liftsto a loop; i.e., the kernel of cb is N. p

B.7. The group G defined above is called the deck group, or group of decktransformat ions.

The deck group depends only on the regular covering, but the isomorphism0 depends on the choice of base points. It is easy to see that if we change basepoints, then we change ds by an inner automorphism.

B.8. We conclude this section with a simple example. Let D° be the puncturedunit disc {zI0 < I z I < 1}. Then 7r,(D°) is infinite cyclic; we choose the generatorto be the loop w(t) = 1/2 e2st, 0 < t < 1.

The universal covering of D° is the upper half plane 112, where the projectionmap is p(C) = e2`K, and the group of deck transformations is the group of trans-lations {lC - C + n; n e Z}.

The covering of D° corresponding to the subgroup Z. is D° itself. Theprojection map is p(C) = C,m, and the deck group is the group of rotations{C - e2i'0lmC' p = 1,..., m}. This deck group has the natural geometric generator

HI.C. Lifting Loops and Regions 45

g(C) = e2i"mC with the property that there is a path w connecting the base pointzo to g(x'0), where p(w) is a simple loop (see II.J.2).

III.C. Lifting Loops and Regions

C.I. Throughout this section, p: 2 - X is some given regular covering, withdefining subgroup N, and group of deck transformations G. Also let zo C-.9, andxo = p(zo) be given base points.

Let w be a loop on X, where the starting point of w is at x'. A spur for w is apath v, starting at xo and ending at x'. Two loops w, and w2 are freely homotopicif there is a spur vm for wm, so that v, w, v;' and v2 w2 v2' are homotopic.

Somewhat more generally, we say that the loop w is freely homotopic to theproduct of loops, fl wm, if there are spurs v and vm, so that v w v-' is homotopicto

C.2. Proposition. If w is freely homotopic to the product [I wm, and each wm liftsto a loop, then w lifts to a loop.

Proof. Since each wm lifts to a loop, so does each vm wm v-'. Hence v w v-' liftsto a loop, so w lifts to a loop.

C.3. Proposition. Let wo, wl, ..., wk be loops on X, where for m > 0, wm lifts toa loop. Let w be freely homotopic to the product wo w...... wk. Then w° lifts to aloop if and only if w$ lifts to a loop.

Proof. Choose spurs v and vm, for w and wm, respectively. Then (v w v-' )' ishomotopic to v w° v-', which is homotopic to { n (vm wm In 7t, (X, x0),this says that [v w v-' ]' = { fl [vm wm v,A' ] }'. Hence in n I (X)/N, where forin > 0, [vm wm um' ] is trivial, [v w v-']2 = [v wo v-')2; i.e., w' lifts to a loopif and only if wg lifts to a loop.

C.4. Proposition. Let D be an open connected subset of X, where every loop in Dlifts to a loop in R. Let 13 be a connected component of p-'(D). Then B is preciselyinvariant under the identity in G.

Proof. Suppose not. Then there are two distinct G-equivalent points x and a' inB. Since they are G-equivalent, p(z) = p(x"). Let w be a path in B from . to z'.Then w = p(iv) is a loop in D that does not lift to a loop.

C.S. The following proposition is essentially immediate from the definitions.

Proposition. Let G be a Kleinian group, where °S = °D/G is a finite (disconnected)Riemann surface. Let F he a full dissection of each component of °S, where r has

46 III. Covenng Spaces

only finitely many elements. Then there is a subcollection F' of curves, so that thefollowing holds. For each connected component Y of S - F', let A be a connectedcomponent of p-' (Y). Let D be the union of all the sets A. Then D is a fundamentaldomain for G.

III.D. Lifting Mappings

D.I. Throughout this section, p: ()?, zo) - (X, xo) and q: (?, $ c,) - (Y, yo) arecoverings with defining subgroups N and M, respectively, and q : (X, xo) -+ (Y, yo)is some continuous map.

A lift of rp is a map 0: (9, go) - (f, Yo) with (p o p = q o ip.Since the maps are all continuous, and preserve base points, there are

induced homomorphisms of fundamental groups:q,: n1(1' Yo) --+ n,(Y yo), and cp,: ni(X,xo) - n1(Y,yo)

Proposition. There is a lift ip of (p if and only if 4p,(N) a M.

Proof. If the lift 0 exists, then for every loop w on 9 at go, ip(w) is again a loop.Hence ep, [ p(w)] a M. Since every element of N can be obtained as the projectionof a loop in 9, cp,(N) c M.

Conversely, assume that ep,(N) c M, and let R be a point of 9. Draw a pathu from .2O to .2`, and define ip(2) to be the endpoint of the lift of qp o p(v), startingat Yo. If fl' is some other path from go to x, then w = v (v')-' is a loop on .9 atgo whose projection determines some element of N; then since ,,(N) a M,cp o p(%) is a loop on Y at yo that lifts to a loop. Hence ep is well defined. Sincethe projections are local homeomorphisms, ip is continuous; it is immediate fromthe definition that cp o p = q o Co. 0

D.2. In the above, note that if cp is a homeomorphism, and qp,(N) = M, then wecan equally well construct the inverse of Co. Hence, in this case, not only do weknow that ip is a homeomorphism of A" onto ?, but we also have shown thattwo points of R project to the same point of X if and only if their images underip project to the same point of Y.

D.3. We continue with the assumptions that cp is a homeomorphism, andep,(N) = M. We assume further that the coverings p and q are regular. Let G bethe group of deck transformations on R, and let H be the group of decktransformations on V.

Let x be some point in the fiber over xo. Then there is an element g e G withg(zo) = R. There is also an element h = ip * (g) e H mapping 9 = 0(20) to Cp(R).

Proposition. 0 * : G -+ H is an isomorphism.

III.D. Lifting Mappings 47

Proof. Since ip is bijective, so is ip *. Notice that for any g e G, Cp o g o co ' is ahomeomorphism of Pthat commutes with the projection q, and Co o g o Co' (y'o) =0og(xo) = iP*(g)o(p(xo) Hence 0*(g) = ipogoCo-1; it follows that CO* is ahomomorphism. Since it is invertible, it is an isomorphism. 0

D.4. We now specialize even further, and assume that 7 = X, and that Y = X.That is, if p: X - X is a regular covering, with defining subgroup N, andcp: X -+ X is a homeomorphism, where cp,(N) = N, then there is a lift ip: Xand there is an automorphism ip*, mapping the deck group G onto itself. Thisautomorphism ip * is well defined only after we have chosen base points in a X,and xo = p(zo)eX. We have assumed that cp(xo) = xo, and we have definedthe lift Co so that ip(zo) = x'o.

Once we have chosen base points, there is a canonical isomorphism 0:n, (X, xo)/N -4 G. Since cp,(N) = N, there is an induced isomorphism rp,,:n t (X, xo )/N -+ n t (X, xo)/N.

Proposition. ip * = 0 o (p o 0-'.

Proof. Let w be a loop on X at xo, and let w be the lift of w to X starting at x`o;call the endpoint z. Denote the class of [w] in n, (X, xo)/N, by the same symbol[w]. Then 0([w]) is the element of G mapping xo to z, and rp* o 0([w]) is theelement of G which maps go to the endpoint of 0(0). Also, [tp(w)],and is the element of G mapping xo to the endpoint of the liftof cp(w), starting at zo. Since ip(w) is the lift of p(w), starting at zo, ip* o 0([w]) =-P o O

D.5. There is one important special case of the above, and that occurs when qPis homotopic to the identity. We actually need a slightly more general statement.

Proposition. Let gyp: X - X be a homeomorphism that is homotopic to the identity.Then there is a lift ip of tp, so that ip commutes with every element of G.

Proof. Let h: X x I X x I be the homotopy, where I is the unit interval. Letv be the path h(xo, t), t e I, and let v be the lift of v, starting at go. For any x e X,let w be a path from z to go; let ip(2) be the endpoint of the lift of 4P o p(w), startingat the endpoint 2' of C. It is clear that ip is well defined, and it is immediate thatpo(p=cpop;i.e.,Cois aliftof cp.

Let g e G, let ip(zo) = ac', and let g(x`o) = Sc. Draw a path w from go to R. Theng o 0(zo) = g(2'), and ip o g(x'o) = 0(2), which is the endpoint of the lift ofp(w), starting at the endpoint of U. Since cp is homotopic to the identity, theloop v cp(w) - v-' is homotopic to w = p(w). Hence the lifts of w v, and v tp(w),starting at zo, have the same endpoint; these endpoints are Cp(2) and g(2'),respectively. 0

48 111. Covering Spaces

III.E. Pairs of Regular Coverings

E.I. Let p: 9 -+ X and q:.9 -. X be regular coverings of the same base X, withdefining subgroups M and N, respectively, where M c N. Then, after choosingbase points, there is also a covering r: (2, go) , (9_ to), where q o r = p. Since Mis normal in n, (X ), it is also normal in N, which we can identify with s, (2).Hence the covering r:9 - X is also regular.

Let G be the group of deck transformations on .9 as a covering of X, andlet J be the group of deck transformations on 9 as a covering of A. Since twopoints of I which project to the same point on A necessarily project to the samepoint on X, J is a subgroup of G. Let H be the group of deck transformationson R.

E.2. Theorem. There is a homomorphism IF: G - H, with kernel J. This homo-morphism is canonically defined by a choice of base point go on A'.

Proof. Choose base points Ro = r(z0), and xo = p(x'o). Let g be some elementof G. Let a2' = rog(zo). Observe that q(:t') = goro9(X0) = Po9(xo) = P(Xo) =x0 = q(90). Hence there is an element +P(g) in H, with !(g)(.to) =1t'; i.e.,W(9)or(X0) = ro9(X0)

Let g, and g2 be elements of G. Then I(g, og2)or(zo) = rog, og2(z0) _'P(91) o r o 92 (xo) _ V'(900 'P(92) o r(zo). Since an element of His determined byits value at any point, P is a homomorphism.

An element g e G lies in the kernel of IF if and only if r o g(zo) = r(zo), whichin turn occurs if and only if g e J. 0E3. We can look at the above from a slightly different point of view. Let p: '° -+ Xbe a regular covering with deck group G, and let J be a subgroup of G. Setk = 2/J, and let r: R - A be the natural projection. If a loop w on A lifts toa loop w on k, starting at some point z, then its lift, starting at some other pointz', is given by some j(w), j e J. Hence if one lift of x is a loop, then every lift of 0is a loop; i.e., the covering r: I - k is regular.

There is also a covering q: A -+ X, with q o r = p. This covering however neednot be regular; it is regular if and only if J is normal in G.

E.4. In general, if p: (2, 20) -. (X, xo) is a covering, then there is a one-to-onecorrespondence between the points in the fiber over x0 and the points in anyother fiber. To see this, draw a path w from go to some point x". Observe that theset of lifts of the projection of this path establishes a one-to-one correspondencebetween the points in the fiber over x0, and the points in the fiber over p(x").

The cardinality of the fiber over any point is called the index of the covering.For a regular covering, the index is equal to the order of the deck group.

E3. We can now state the following corollary to E.2.

III.F. Branched Regular Coverings 49

Proposition. The order of J is equal to the index of the covering r: 9 -+ 8, andthe index of J in G is equal to the index of the covering q: X -+ X.

III.F. Branched Regular Coverings

F.I. A freely acting discontinuous group on 9 makes R` into a regular coveringspace; if G acts discontinuously, but not freely, then we get a branched regularcovering. For our purposes, it suffices to restrict attention to dimension 2, wherethe topology is easier. We could also define branched coverings that are notregular, but we have no need of these more general objects.

Throughout this section, all spaces are manifolds of dimension 2.

F.2. In order to define a branched regular covering, we make the followingsuppositions. We are given a space 9, and a group G of homeomorphismsacting on 9. Set X = A`/G, and let p: 2 -. X be the natural projection; weendow X with the usual identification topology, so that p is both open andcontinuous. Assume that G acts freely except at a discrete set of points {Xm};set 01 = I - {zn, }, and set °X = p(°I). The map p: )? -+ X is a branched regularcovering with branch points {(2m)}, special or ramification points {xm}, wherexm = p(zm), covering space 1, base X, and projection p, if the following hold.

(i) p: °A` -4 °X is a regular covering with deck group G;(ii) For every branch point z, Stab(z) is finite, and there is a topological disc

U containing z, called a nice neighborhood of x, so that U is precisely invariantunder Stab(z) in G.

F3. For each branch point z, the branch number, or order at z is the order v ofStab(z). This number v is also called the ramification number or order at x = p(x").

F.4. Let 0 be a nice neighborhood of the branch point z, and let U = p(ly).Let 00 = 0 fl 09, and °U = U fl °X. Then p: 00 -+ °U is a regular covering of °U,with finite deck group Stab(z). Since 0 is a topological disc, °U is a punctureddisc, and hence, so is °U. This covering is equivalent to one of the coverings ofa punctured disc, of finite index, mentioned in B.8. Using this equivalence, thereis a well defined geometric generator g for Stab(x'); there is a path w in °0 joiningsome point z' to g(2'), and w = p(Q) is called a small loop about x = p(2). Notethat w" lifts to a simple loop on 00.

F3. Pick a base point go on °a1, and let N be the defining (normal) subgroup forthe (regular) covering p: (°1, 2,) - (°X, xo), where xo = p(x'o).

Pick a small loop wm about each special point xm, and let v, be a spur towm. Let vm be the ramification number at xm. Then [wm] = [vm (wm)"^' v.-'] _[vm.wm.vm']"-'eN.


Let No be the smallest normal subgroup of n,(°X,xo) containing all theseelements [wm]. Observe that No depends only on the points xm and their ramifica-tion numbers; it is independent of the choices of the particular small loops, or ofthe spurs.

If No = N, then we say that p: )7 -. X is the branched universal covering ofX, branched at the points {xm } to order {vm}; we usually shorten this description,and simply say that p: 2 - X is the branched universal covering.

F.6. Theorem. Let p: 2 -, X be a branched regular covering. A loop w on ° ishomotopically trivial in 2 if and only if [p(w)] a No.

Proof. If w is homotopically trivial in R, then since 9 - °2 is a discrete set ofpoints, w is freely homotopic in °J2 to a product of small loops about these branchpoints. The projections of small loops about the branch points are precisely theloops that generate No.

Conversely, if w is a loop in °X, where [w] a No, then w is freely homotopicto a product ll(vm)°", where each vm is a small loop about a special point, andeach am is a multiple of the ramification number at that point. Hence each (vmr"lifts to a power of a small loop about a branch point; such a loop is of coursehomotopically trivial in 9.

F.7. Corollary. A branched regular covering p: ` - X is the branched universalcovering if and only if 2 is simply connected.

Proof. If 9 is simply connected, then by F.6 every loop on °2 projects to a loopin No. Conversely, if p: 9 - X is the branched universal covering, then No isthe defining subgroup for the covering p: °2 'X; hence by F.6, every loop in°)7 is homotopically trivial in 9. Since 9 - °2 is a discrete set of points, R issimply connected.

F.8. Proposition. Let q: X -* X be a branched regular covering. Then there is abranched universal covering p: )7 -i X, and there is a regular covering r:so that p=qor.

Proof. Choose base points .9Q a °2, and xo = p(Ao). Let p: (2, go) - (X, x0) bethe branched universal covering, where p and q are branched over the samepoints, and are branched to the same order over each of these points. Thenp: °2 - °X, and q: °X °X are both regular coverings with defining subgroupsN and M, respectively. Since a small loop about a special point in X, when raisedto a certain power, lifts to a loop in both °Jl and °$, and N is normally generatedby these loops, N c M. Thus there is a covering r: °2 -+ °tS, with p = q or.

Let z be a branch point of order v on X, and let w be a small loop about z.Set w' = p(w), and observe that there is a small loop w about p(z), so that

M.G. Exorcises 51

[w'] = [w°]. We have defined the covering p so that w" lifts to a loop on X, andv is the smallest positive integer for which this is true. We conclude that r(w) isa small loop about some branch point in X, we define it to be r(x'). Note that rmaps a nice neighborhood of x homeomorphically onto a nice neighborhood ofr(z). It follows that r: X - X is a covering. Since A' is simply connected, it is aregular covering.

F.9. Not every regular covering p: °a1 - °X extends to a branched regular cover-ing. Again, let {xm} be a discrete set of points of X; set °X = X - {xm}. Let wmbe a small loop about xm, and let vm be a spur connecting the base point of wmto some base point x0.

Proposition. Let p: °R` - °X be a regular covering. Then °X can be embedded ina space X, and p can be extended to a branched regular covering p: X - X if andonly if there are positive integers vm so that for every m, vm (wm) . v,,' lies in N,the defining subgroup for this covering.

Proof. We already know that the statement is true if p extends; that is, if p isalready defined as a branched regular covering, p: X -. X.

Conversely, suppose each of the loops wm, when raised to some power, liftsto a loop. Let U. be a neighborhood of xm, where U. - {xm} is topologicallya punctured disc. Then each connected component V.,,, of the inverse imageof Um, is precisely invariant under a (possibly trivial) finite cyclic subgroup ofthe deck group; if vm is minimal, then this subgroup will have order vm. Thenp1Q.,, is a covering, of finite index, of Um. By B.8, V... is homeomorphic tothe punctured disc. Hence we can rill in the missing point JCm,, over xm.

III.G. Exercises

G.I. Give an example of a continuous surjection which is a local homeomorphism,but not a covering.

G.2. If p: X - X is a covering of a torus of finite index, then X is also a torus.

G.3. Consider the function e= as a mapping of C = C - {0} to itself. To whichcoverings of C can this mapping be lifted.

G.4. Find all branched regular coverings of the 2-sphere that are branched overat most one point; two points.

G.5. Let G be the Kleinian group generated by g(z) = l/z. Then p: Q(G) - Q(G)/Gis a branched regular covering. Describe this covering; i.e., find Q(G), find the


branch points, describe Q/G topologically, find the number and orders of theramification points. Also find an explicit formula for the projection map.

G.6. Same as G.5, but for the group generated by z - 1/z and z - -z.

G.7. Same as G.5, but for the group generated by z - 1/z and z e2i imnz, n z 3.

G.B. Same as G.5, but for the group generated by z -+ z + I and z -' - z.

Chapter IV. Groups of Isometries

In this chapter we describe the three basic geometries, elliptic, parabolic, andprimarily hyperbolic, that are important for the theory of Kleinian groups. Wethen build some of the theory of discrete groups of isometries in these geometries.The major results are the construction of the Dirichlet and Ford regions, andthe proof of Poincare's polyhedron theorem.

W.A. The Basic Spaces and their Groups

A.I. We denote Euclidean n-space by E"; if it is necessary to use coordinates,we will write a point x e iE" as x = (x 1, ... , x"), or sometimes as (x, t), combiningthe first n - I coordinates as x, and distinguishing the last coordinate as t. Whenwe write a point in iE" as (x, t), we refer to the t coordinate as the height, so thatfor example, the line {(0,t)} is a vertical line.

In F", the unit sphere is S"-' = {x I I x I = 1), the open unit ball is B" _{xIIxI < 1}, and the upper half space, H" = ((x, t)It > 0).

The one point compactification of F" is denoted by E"; the added point is ofcourse called oo.

The natural inclusion of Gr` in E", given by F"-' _ {(x, t) It = 0), extends tothe one point compactifications, so that E"-' = OH", the (Euclidean) boundaryof 0.0".

There is also a natural inclusion of U-0" into H"+' defined by 9-U" _{(x,t)a0-8"+'IxeIE"-'}.

Unless specifically stated otherwise, we will assume throughout this chapterthat n >- 2.

A.2. Each of these spaces has a natural metric on it. The Euclidean and sphericalmetrics need no further mention, but we do need to identify their groups ofisometries. The group of isometrics of S"' is the orthogonal group; as usual, wedenote it by O. The group A" of isometries of IF" is generated by 0" and theEuclidean translations.

A3. The local differential metric on I--0" is given by

54 IV. Groups of Isometrics

ds2 = (dx2 + dt2)/t2.

With this metric, which is also called the Poincare metric, H" is a model ofhyperbolic n-space.

We define a group of motions L" of t" by writing down a set of generators.For the first two sets of generators, we write a point in 0=" as (x, t).

(i) Translations: (x, t) - (x + a, t), a e F"-1; oo - oo.(ii) Rotations: (x, t) - (r(x), t), r e ®n-1; oo - o0For the remainder of the generators, we write a point in L" simply as x.(iii) Dilations: x Ax, A > 0, A # 1; oo - oo.(iv) Inversion: x - x/1x12; 0 - oo; ox - 0.It is clear that every element of L" preserves both H" and its boundary, E"-1

Proposition. L" acts as a group of isometries on H".

Proof. The translations and rotations offer no difficulty, for they preserve dx2 +dt2, and they preserve the height t.

For a dilation g = (g1,...,g"), where g(x) = Ax,

dg = A2 Y dxM,

hence

9"2dg,=xn2dx"2,.For inversion, f = (f1, ... , f"), we compute

Olm/t3xk = 1x1-25km - 2xkxmlxl-4

(as usual, bk",= 1,ifk=m,andbk",=0,ifk :0 m). Then

Y-afm -afm = Ixl_8

2xmxj)(IXI2bkm - 2xkxm),,, aXj l3Xk m

=IXI_8

(IXI4amJakm - 2bjmXA:x.lx 2 - 2dkmXmxjIXI2 +

Ix1-8(IX14bjk - 2xjxkIX12 - 2xjxkIxI2 + 4XjXk E xm)

=Ixl-4ajk.

An easy computation now yields

(dfm)2 = IXI-4 Y (dxm)2.

The group L" generated by the elements (i) through (iv) above is sometimescalled the (n - 1)-dimensional Mobius group. We will see in B.7, that L" is thefull group of isometrics of H".

A.4. In its action on P (or ?-1), L" acts as a group of conformal motions, butnot as a group of isometries in any metric.

I V.A. The Basic Spaces and their Groups 55

For our purposes, a differentiable homeomorphism f of Euclidean space isconformal if at each point, the differential Df is a scalar multiple of an orthogonalmatrix. It is an exercise to show that f is conformal if and only if Df preservesthe (unoriented) angle between any two tangent vectors. We do not requireconformal maps to preserve orientation.

We use inversion j to define local coordinates near oo. A map f is conformalat oo if f o j is conformal at 0; Similarly, f is conformal at f '(00) if j of is.

Translations and rotations are Euclidean isometrics, so of course they areconformal on E". If f is a dilation, then Df is a constant multiple of the identity.We showed above that if j is inversion, then Dj is a multiple of an orthogonalmatrix; hence inversion is conformal in P. We use inversion to define localcoordinates at cc, hence inversion is automatically conformal at 0 and oo.Inversion commutes with rotations and conjugates a dilation into its inversedilation, so rotations and dilations are conformal at oc.

The only thing left to check is that translations are conformal at oo. Letg(x,t) = (x + a, t), a = (a1,... , a,-,), be a translation, and let j(x) = xll x 12 beinversion. It obviously suffices to assume that all but one of the a. = 0; we assumeak # 0. Let P,, be the 2-plane spanned by the coordinates xk and t (i.e., all theother coordinates are zero). Then g and j both keep Pk invariant. In Pk, setC = xk + it, and observe that j o g o j l Pk(t) = C/(ak C + 1), which has derivative 1at the origin. This mapping acts as the identity in the other coordinate directions;hence the differential of j o g o j at the origin is the identity.

A.S. There are several embeddings of L" into L"', and there is one naturalembedding. For the moment, we only show that for each g e L", there is a g' c- Ln+'so that g'lt" = g, and g'(0-0") = 0-0"; the proof that this defines an isomorphicembedding can be found in A. 11. For our present purpose, it suffices to showthat there is such an extension for each generator g of L". If g is the trans-lation: (x, + a,,...,x"_, + a"_,,t), then g' is the translation:(x1,...,x",t)-+(X, + a1,...,x"_, + a"_,,x", t). If g is the rotation: (x1,...,x"_t) (r1(x),...,r"_,(x),t), then g' is the rotation: (x1,...,x",t) _+(r1,...,r"_t,x", t).If g is the dilation: x - Ax, then g' is the corresponding dilation x -* Ax in L"+'Similarly, if g is inversion, then so is g'.

It is important to observe that this extension can be viewed in two ways.First, we can view the n coordinates of E" as being the first n coordinates of E"+';in this view H" lies in the boundary of H". One can also view the n coordinatesof E" as being all but the penultimate coordinate of E"+'; in this view, H" c 00"'Both

of these embeddings of E" into E"+t yield the same extension of L" into L"'.

A.6. We regard L3 as acting on V, where it acts as a group of conformal motions.The transformations (i) through (iv) acting on C = P are just translations,rotations about the origin, reflections in lines through the origin, dilations, andinversion in the unit sphere. Using I.C.4, it is easy to see that L3 = IVA, the groupof all conformal mappings of C, including those which reverse orientation.


A.7. Rather than introduce more notation, we remark here that k" is both thefull group of isometries of I-I", and also the full group of conformal maps of E"-1.With one exception, the proofs of these facts will be spread out over this andthe next section. Thus far we have shown that every element of 1L" is both anisometry of l-0", and a conformal homeomorphism of t"'. We will show in B.7that every isometry of I-l" is in d"; it is immediate that an isometry of H" can haveat most one continuous extension to a homeomorphism of P`; we will see inA. 11 that distinct elements of L" have distinct actions on ltn-1. Since we will makeno use of either the fact or the proof, we do not include a proof of Liouville'stheorem that every (local) conformal homeomorphism of E"-1 lies in I_"

A.S. As in dimension 2, we regard Euclidean k-planes in I:" as k-spheres in E".

Proposition. The elements of I_" preserve the family of k-spheres in t", k = 0,I,...,n-2.

Proof. It suffices to consider the case k = n - 2, for lower dimensional spheresare intersections of higher dimensional ones. Recall that codimension one spheresare called hyperspheres.

The Euclidean motions in Q_" are Euclidean motions of E", so they preserveboth hyperplanes and hyperspheres. Easy computations show that the equationsfor a hyperplane,

F, a",x,, = b,

and for a hypersphere,2= 2

are carried over by a dilation into expressions of the same form.A hyperplane through the origin, 0, is mapped by inversion onto

itself. Any other hyperplane, Y- a",x,,, = b, b * 0, is mapped by inversion onto thehypersphere b>x"2, - 0. If the hypersphere Y(x, - a.)2 = r2 passesthrough the origin (i.e., if Ea. = r2), then this hypersphere is mapped byinversion onto the hyperplane 1/2; otherwise it is mapped onto thehypersphere (r2 - 2Za",x," - 1 = 0. 0

A.9. A nested set of spheres in E"-` is a set of spheres So c S, c . c S"_2i whereS. is a sphere of dimension m. Each S. divides Sin+t into two discs; the nested setof spheres is oriented if, for each m, one of these discs is chosen as the positivehalf; S"_2 divides S"_t = En-1 into two discs, we require that one of them bechosen as positive; we also require that one of the two points of So be chosen asthe positive half.

Proposition. Given two oriented nests of spheres So c c S"_2, and To c cTn_2 in t"-`, there is an element ge0_" mapping one nest onto the other (i.e.,

I V.A. The Basic Spaces and their Groups 57

g(SS) = Tm, and g maps the positive half of S. - S.-1 onto the positive half ofT. - Tm_1). Further, if x is any point on the positive half of S, - S°, then g(x)can be chosen arbitrarily on the positive half of T, - T°; g is unique once this choiceis made.

Proof. It suffices to assume that To c c T"_2 is the standard nest to e . . . CI'"-2, where to = {0, oo}, with 0 the positive half, and the positive half oftm+1 - Pm is H". Further, we choose the point I on I-I', and require thatg(x) = 1.

We first assume that n >- 3.If S"_2 is a Euclidean hypersphere, translate it so that it passes through 0,

then inversion maps it to a hyperplane, a translation maps this to a hyperplanethrough the origin, and a rotation then maps this hyperplane to t". If thepositive half of E"' - S"_2 has not been mapped to H", then follow thissequence of maps with the rotation (x, t) -+ (x, - t), x e E".

If Sn_2 is a Euclidean hyperplane, then pick up the above sequence at thethird step. We have produced a motion g, a I_" so that 9, M-2) and g,maps the positive half of t` - Sn_2 to H".

Exactly as above, there is a motion 92 a O" mapping g, (Sn_3) ontowhile mapping the positive half of g,(Sn_2) - g1(Sn_3) onto H"-2. Now use theextension given in A.5, so that we can regard g2 as an element of I_". Then g2 ° gtmaps Sn_2 to E"-2, Sn_3 to !fin-3, the positive half of t"-' - Sn_2 to H"-', andthe positive half of Sn_2 - S"_3 to H"-2.

We continue as above until we reach dimension 2; that is, we have foundg' c- L", where g' maps Sk to Ik for k >- 2, and for k > 2, g' maps the positivehalf of St - Sk_, to BIk. In E2 = C, it is routine to find a fractional lineartransformation h' mapping g'(S0) to {0, co}, with the positive half mapped to 0,and x to 1. This necessarily maps g'(S,) onto V; if necessary, we follow thisdirectly conformal map with the map z -> z so as to obtain h e I.3 mapping thepositive half of 9'(S2) - g'(S,) onto H2. Use A.5 to extend h to I_", and observethat hog' is the desired map.

For n = 2, it is standard to write down an element of I.2 that maps a tripleof distinct points onto {0, 00, 1).

Now suppose that g keeps the standard oriented nest invariant. If n = 2, theng e I.2, g(0,1, oo) = (0,1, oo), and g preserves orientation. If n >- 3, consider thedifferential Dg(0) at the origin, and look at its action on the standard orthogonalframe. Since g preserves E', the first basis vector, (1,0,. .. , 0) is an eigenvectorof Dg(0), with positive eigenvalue. Similarly, g preserves I:', 1E2, and 112, andso the second basis vector, (0, 1,0,. .. , 0) is also an eigenvector of Dg(0), withpositive eigenvalue. Continuing in this manner, we see that Dg(0) is diagonal,with positive entries; since g is conformal, Dg(0) is a positive multiple of theidentity.

Since g(0) = 0, and g(oo) = oo, g preserves the family of lines through theorigin. Infinitesimally, at 0, each of these lines is mapped onto itself; hence


g preserves each line in this family. Since Dg(0) is a positive multiple of theidentity, g preserves each ray of each of these lines. The family of orthogonaltrajectories to the above family is the family of hyperspheres centered at 0; sinceg is conformal, it also preserves this family. Since g(1,0,. .. , 0) = (1, 0, ..., 0),g preserves §n-2; since g preserves each ray through the origin, glS"-2 is the

identity.Consider a 2-plane P passing through 0, oo, and (1, 0,..., 0). Now g(P) is a

2-sphere passing through 0 and oo; hence it is a 2-plane. The circle p fl §"-2 ispointwise fixed by g; hence g(P) = P. Since gI P is conformal, has a circle of fixedpoints and two additional fixed points; it is the identity. We conclude that g isthe identity on every line through 0; i.e., g = 1.

A.10. Proposition. Let S be a hypersphere in t"-'. Then there is a unique reflectiong e L", where giS = 1, and g interchanges the two halves of t"-' - S.

Proof. The transformation (x, t) - (x, - t) is a reflection in t". By A.9, there isan element of L" mapping S onto f"-2. Hence there is a reflection in S. Theuniqueness follows from the uniqueness statement in A.9, by choosing someoriented nest where S = Sn_2.

It follows at once from the uniqueness of the reflection, that if j denotesreflection in t"-', then j commutes with every element of L", regarded as amapping of E".

A.11. We are now in a position to show that the extension of A.5 defines anisomorphic embedding of L" into L"+'. For each generator g e L", we have defineda g' e L"+' so that g' J P = g, and so that g' preserves H". Using composition, weobtain the same result for any g e V. It is immediate that if g # 1, then g' # 1.If g = 1, then g' acts as the identity on P, and preserves H"'. By the uniquenessstatement of A.9, g' = 1.

A.12. Proposition. The stabilizer of H" in L"+' is V.

Proof. Denote the stabilizer of H" in L"+' by G. It is immediate that L" a G. Ifg e G, then by A.9, there is an element h e L" so that hl t"-' = g l P-'. Thenk = go h-' is the identity on t", and preserves H"; by the uniqueness part ofA.9, k is the identity on P. We conclude that k is the identity as an element ofL"+,

A.13. It is easy to see that L" is generated by reflections. If g e L", viewed as actingon t", is the reflection in the (n - 1)-sphere S._t, then there is a unique n-sphereS. in P+', where S. passes through S._1 and is orthogonal to P. The extensiong' of g is the reflection in S,,.

IV. B. Hyperbolic Geometry 59

There is another embedding of L" into L"+' that is worth mentioning. Thestandard embedding of HO into 1H"+' is given" by (x 1, .-.,x,-,, t)-'(x1,. .. , xn_1, 0, t).Extend the embedded H" to a Euclidean sphere S c E"+' Every g E L", as aconformal map of S, has an extension g": E"+` -. En. Since S is orthogonal to8H"+1, #(H"+1) = H"+1 By A.9, there are exactly two choices for g; one choicepreserves each half of FD"+' - FI" and preserves parity (that is, g and g either bothpreserve or both reverse orientation), and the other choice interchanges thetwo halves of H"+' - FI" and reverses parity. Notice that both choices give usan embedding of L" into L"'. The parity preserving embedding is the standardone.

From the point of view of intrinsic geometry, the natural way to view theembedding of L" into L"' is given above. From the point of view of Mobiustransformations, the natural point of view is as follows. Start with an elementg e L", acting as a conformal map of "; then regard E" as the boundary of FI"+',so that there is a unique element g of L"', with boundary values g; g is the imageof g under the natural embedding.

IV.B. Hyperbolic Geometry

B.I. The two standard models for hyperbolic n-space are FI" and B". Even thoughA.9 assures us that there is a homeomorphism in L"+' mapping one onto theother, it is sometimes convenient to have an explicit map between them.

Our map, which is slightly more complicated than necessary, is as follows.We start with the dilation x - x/2, then the translation, (x, t) - (x, t - 1/2), thenwe invert in the unit sphere, so that the composed image of B" is the half-spacet > 1. Follow this with the translation (x, t) - (x, t - 1), so that we now have amap from B" onto FI". Call the composed map q-'.

Observe that q'' restricted to S"', the boundary of B", is a conformalmap of S"' onto t"'. If we compose q-' with the mapping (x, t) - (x, - t), thenwe obtain a variation of the usual stereographic projection. In this projection,the plane intersects the sphere at the equator. Otherwise the projection is theusual one; that is, one draws the line L from the north pole, (0, 1) to a pointx e S"; then q'' (x) is the point where L intersects lE"'1.One could equally welldefine q by drawing the line L from a point x e E"-' to (0, 1); then q(x) is the pointof intersection of L with S"-' (of course, q(oo) = (0,1)).

B.2. The group of hyperbolic isometries of B" is defined by P" = qL"q-'. We willalso sometimes refer to P" as FI" normalized so as to act on B".

We know from A.12 that L" is the stability subgroup of H" in L"+'; it followsthat P" is the stability subgroup of B" in L"+'. An important application of thisremark is that 0" c P".


The inclusion of L" into 0_"+' given by A.5, 11 carries over to an inclusion ofP" into P"+I

B.3. Proposition. P" acts transitively on B".

Proof. We already know that the orthogonal group, ®" c P"; hence it sufficesto show that we can map any point of the form (x, 0,..., 0), x > 0, to theorigin. For n = 1, the transformation g(t) = (t - x)/(l - xt) accomplishes this.The extension of this transformation from P' to P" preserves B", and maps(x, 0,..., 0) to the origin. p

As in dimension 2, P" in fact acts transitively on the tangent space to B".This follows from the fact that 0" acts transitively on the set of directions at 0,so we can not only map x to 0, but we can also map an arbitrary direction at xto an arbitrary direction at 0.

B.4. Proposition. In H", the geodesics are the arcs of circles orthogonal to theboundary, t"-'.

Proof. Let x and x' be points of H"; using transitivity of points, we can assumethat x = (0,...,0, 1), and then using transitivity of directions, we can assumethat x' lies on the same vertical line as x, and that x' is higher than x; thatis, x' = (0,..., 0, a) a > 1. If v(s) = (vl (s), ... , v"(s)) is any path from x to x',parameterized by the unit interval, then the hyperbolic length of v is

f0t

(Y(dvm/ds)z)tn 1 I dv"ds.

v"(s)

('ds >_

o v"(s) Ws

The expression on the right is the hyperbolic length of the Euclidean linejoining x to x'. Equality in the above holds only if dv,"/ds = 0, for every m # n;i.e., equality holds only if v is this straight line. We conclude that for generalx and x', there is a unique geodesic between x and x'; it is the arc between x andx' of the unique (Euclidean) circle passing through these points, and orthogonalto 00-0". p

B.S. An easy computation shows that for x and x', normalized as above, thehyperbolic distance d(x, x') = log(a).

Note that 0-I" is complete in this metric; in particular, a subset of H" is compactif and only if it is closed and bounded. The points of OH" = 1'"-' are infinitelyfar from any interior point; is called the sphere at infinity of the hyperbolicspace.

IV. B. Hyperbolic Geometry 61

B.6. In hyperbolic space (that is, H" or B"), we regard Euclidean circles orthogonalto the boundary as being lines. Similarly, the Euclidean k-spheres orthogonalto the boundary are k-planes. Where there is some danger of confusion, we willidentify these as hyperbolic lines or planes.

At the origin in B", the two notions of k-plane coincide. Hence the planes inhyperbolic geometry have the same incidence relations as those in Euclideangeometry. Also, since the two geometries are conformally the same, any truestatement about planes and angles at a point in Euclidean geometry is also truein hyperbolic geometry. One example of this is the following.

Proposition. Let x be a point of H", and let L be a k-dimensional subspace of thetangent space at x. Then there is a unique (hyperbolic) k-plane through x whosetangent space at x is L.

B.7. Theorem. 1_" is the full group of isometries of 0-0".

Proof. Let g be some isometry of B". If g(O) # 0, then there is an element f, e IP"with f, o g(O) = 0. Since f, o g is an isometry, it preserves every hyperspherecentered at the origin, from which it follows that its differential at the originpreserves the length of every tangent vector; i.e., D(f, o g)(0) is an orthogonaltransformation. Then there is an element f2 a O" with D(f2 of, o g)(0) = 1. Theisometry f2 of, o g preserves every (Euclidean) ray emanating from the origin,for these are hyperbolic geodesics. Since f2 of, o g preserves every ray emanatingfrom the origin, and preserves every sphere centered at the origin, it is theidentity. 0B.B. Corollary. The stabilizer of the origin in P" is 0".

B.9. Corollary. Let g e !" be such that g(0) = 0, and Dg(0) = 1, then g = 1.

B.10. Proposition. Let L be a hyperbolic hyperplane. Then there is a uniquereflection g in L (i.e., g is a hyperbolic isometry, g interchanges the two half spacesbounded by L, and g I L = 1).

Proof. Regard the hyperbolic space as H", complete L to a hypersphere in s",and let g be the reflection in this hypersphere. Since L and 80.0" are orthogonal,reflection in L keeps 1-I" invariant. Hence g e V. If g' is another reflection inL, then for any point x e L, g-' o g'(x) = x, and D(g-' o g')(x) = 1; hence g = g'.

0

&11. If x # y are points of hyperbolic space, then the perpendicular bisector ofthe line segment between x and y is a hyperplane.


Proposition. Let x # y be points of hyperbolic space, and let L be the perpendicularbisector of the line segment joining x to y. L divides hyperbolic space into two halfspaces; let H be the half space containing x. Then H = {zI d(z, x) < d(z, y)}.

Proof. Reflection in L keeps every point of L fixed and interchanges x and y.Hence every point of L is equidistant from x and y.

Assume there is a point z in H equidistant from x and y. Let w be the pointwhere the line segment between z and y crosses L. Then d(x, z) = d(y, z) =d (y, w) + d(w, z) = d (x, w) + d(w, z). Since z e H, w does not lie on the line betweenx and z, contradicting the fact that there is a unique geodesic connecting x to z.No point of H is equidistant from x and y, so by continuity, no point of H canbe closer to y than to x.

B.12. A sphere is of course the set of points at a given distance from a given point.

Proposition. In H" (or in P"), hyperbolic spheres are Euclidean spheres (in general,with different centers and radii).

Proof. Let S be a hyperbolic sphere centered at 0 e B". Since 0" c P", and 0"acts transitively on the directions at 0, S is also a Euclidean sphere. A spherecentered at any other point is just g(S), for some g e P. By A.8, g(S) is a spherein E". Since g(S) does not pass through oo, it is a Euclidean sphere.

W.C. Classification of Elements of U"

C.I. The classification of the elements of L" is very similar to that of I.B; only thetransformations here are somewhat more complicated.

Every g e !L" has at least one fixed point in the closure of H". If g has a fixedpoint in H", then it is elliptic; if g is not elliptic, and has exactly one fixed pointon aw, then it is parabolic; otherwise it is loxodromic.

C.2. If g is elliptic, then normalize so that g acts on B", and so that a fixed pointof g is at 0. Then by B.8, g e 0". We conclude that every elliptic element of L" isconjugate in I."' to an element of 0".

Regard g e 0" as a linear transformation acting on E". The set of fixed pointsof g is the nullspace of I - g. In B", this is both a Euclidean and a hyperbolick-plane, for some k < n. Therefore, the fixed point set of a general elliptic elementof L" is a hyperbolic k-plane.

For n odd, every element of 0" has at least one eigenvalue = ± 1. For n = 3,the case we are most interested in, an orientation preserving non-trivial elementof 0' has exactly one fixed direction. This means that an elliptic element g e Mhas a unique hyperbolic line of fixed points in H3; this line is the axis of g. The

W.C. Classification of Elements of L 63

endpoints of the axis are the fixed points of g on C; one easily sees that this is anelliptic element of M.

C3. Now assume that g e U' is not elliptic.If g has three distinct fixed points, xt, x2, x3, in t"', then consider the circle

C determined by these three points. Since g fixes three points on C, it keeps Cinvariant. Further, by A.9, gIC = 1. Next let P be the 2-plane in H" whoseboundary is C; observe that since g(C) = C, g(P) = P. Complete P to a Euclidean2-sphere, and use A.9 again to conclude that g I P = 1. We have shown that if ghas three distinct fixed points in OH", then g is elliptic.

C.4. Using A.9, we can map any pair of distinct points on the sphere at infinityto any other such pair. There is clearly a unique line between 0 and oo; hencethere is a unique line between any pair of distinct points on the sphere at infinity.

Suppose that g E I_" is loxodromic, with fixed points z and z'. Let A be theline in H" joining z to z'; A is the axis of g. Conjugate g by an element of I." sothat A is the vertical line {(0, t)I t > 0}; that is, after conjugation, the fixed pointsof g on the sphere at infinity are at 0 and co.

Pick a point x on the axis A of g; then g(x) also lies on A. There is a dilationd, with d o g(x) = x, hence d o g I A = 1. Since D(d o g) (0) is a multiple of an orthog-onal transformation, and is the identity in one direction, it is orthogonal. Weconclude that there is an orthogonal transformation r, where r fixes the verticaldirection, so that r o d o g(0) = 0, and D(r o d o g)(0) = 1. Then g' = r o d o g keepsI-0" invariant, and pointwise fixes A. For any (n - 1)-sphere S, centered at 0, g'(S)is an (n - 1)-sphere centered at 0 (the reflection in g'(S) interchanges 0 and co),of the same Euclidean radius (S and g'(S) both intersect A at the same fixed pointof g'). Since g' keeps B" invariant, g' e P"; by B.9, g' = 1. We conclude thatg-t = rod; that is, g is a rotation followed by a dilation. Since rotations anddilations with the same fixed points on the sphere at infinity commute, we canalso regard gas a dilation followed by a rotation. Of course if d is trivial, then gis not loxodromic but elliptic.

CS. As in dimension 2, a loxodromic transformation which is conjugate to adilation is called hyperbolic. The dilation x A.2x can be realized as the productof two reflections: first, reflect in the unit sphere, then reflect in the sphere I x I = 7.

Conversely, if g is defined as the product of two reflections, in disjoint spheres,then every circle orthogonal to the two spheres is invariant under g; hence g ishyperbolic.

C.6. The last case is that g is parabolic. We normalize so that the fixed point ofg is at co, and we regard g as acting on OH" - {co} = E"'. Then there is atranslation f so that f o g(0) = 0; i.e., fog is loxodromic or elliptic with fixedpoints at 0 and oo. Then fog = rod, where r e O"-t, and d is a dilation or theidentity.


Write g(x) = r o d(x) + b, and use the contracting mapping principle to observethat since g has no finite fixed point, d must be trivial. Also, regarding r asa linear transformation, and b as a vector, b is not in the range of 1 - r. Therange and nullspace of 1 - r are orthogonal, so there are vectors y, and y2with b = g(O) = (r - 1)(y,) + y2, where (r - 1)(y2) = 0. Now conjugate g bythe translation x - x + y,; after this conjugation r(b) = b. Next, conjugate bya dilation so that IbI = 1, and, finally, conjugate by a rotation to obtain thenormal form for a parabolic transformation: g(x) = r(x) + b, where b = (0, 1) andr(b) = b.

C.7. We have shown that if g e L" is non-trivial, then the fixed point set of g, inits action on (?", is either empty, one point, or a sphere of dimension at mostn - I (n - 2 if g is orientation preserving).

As in dimension 2, if g e 1_" has finite order, then g is elliptic, and if g is anelliptic element of a discrete subgroup of L", then g has finite order.

C.8. As in dimension 2, most elements of L" keep no ball in I"-' invariant.If g is elliptic, then we can assume that it is a rotation acting on B". If g keeps

some Euclidean ball on S"' invariant, then g must fix the center of the ball;i.e., g has an eigenvalue equal to + 1. Conversely, if I - g is singular, then ghas at least one pair of antipodal fixed points on the sphere at infinity; everyball on the sphere at infinity, which has one of these points as center, is also keptinvariant by g. Therefore, in this case, except perhaps for these two fixed pointsof g, every point on the sphere at infinity lies on the boundary of an invariantball.

If g is loxodromic, then we can assume that the fixed points of g are at 0 andoo in E"-', and that g = do r, as above. It is clear that the fixed points lie onthe boundary of every invariant ball; also, d keeps every hyperplane throughthe origin invariant. Therefore g keeps a ball in l"-' invariant if and only ifthe ball is a Euclidean half-space kept invariant by r; r keeps such a half-spaceinvariant if and only if it fixes the direction orthogonal to the boundary hyper-plane. We have shown that g keeps a ball invariant if and only if + 1 is aneigenvalue of r. In this case, the boundary of the invariant hyperplane might beunique.

If g is parabolic, then we normalize so that g(x) = r(x) + b, where b = (0, 1)and r(b) = b. Every invariant ball contains oo in its closure, so is a half space.Also, every invariant ball contains a vertical line.

If r = 1, then every hyperplane containing a vertical line is invariant. If r # 1,then the invariant half spaces come in parallel families: for each family, thecommon normal to the boundary lies in the null space of I - r, and is orthogonalto the vertical direction. In this case again, if g has one invariant ball, then forevery non-fixed point x on the sphere at infinity, there is an invariant ball whoseboundary passes through x.

IV-D. Convex Sets 65

IV.D. Convex Sets

D.I. In this section, we restrict our attention to hyperbolic 3-space, X03, and itsboundary V. For a set Y c H3, the convex hull of Y is simply the smallest convexset containing Y. For a set y c E2, the convex hull, K(Y) is the intersection of allthe closed half spaces in H3 whose Euclidean boundary, on the sphere at infinity,contains Y

If G is a Kleinian group, then the associated convex region K = K(G) is theconvex hull of the limit set, A (G).

D.2. Let W be a simple closed curve in S2. Then W divides §2 into two closedtopological discs; we label one of these discs as the inside disc, B and the otheras the outside disc, B2. Then K(B1) fl K(B2) = K(W). The boundary in 033 ofK(B1) is called the outside boundary of K (W), and the boundary of K(B2) is calledthe inside boundary of K(W). The reason for this terminology is that a path in033 starting in the inside disc and ending in the outside disc enters K(W) throughthe inside boundary, and leaves K(W) through the outside boundary. To see this,normalize so that W is a bounded curve in V. Then the inside boundary (thatis, the boundary of the convex hull of the outside region) is defined by hyperbolicplanes whose boundary circles lie inside or on W; Similarly, the outside boundaryis defined by hyperbolic planes whose boundary circles lie outside or on W.

If W is a circle, then K(W) is the plane whose Euclidean boundary is W In thiscase, the inside and outside boundaries coincide.

Let H be a closed half space in 0-3, where H contains K(B2); that is, theEuclidean boundary of H does not intersect 92. In H3, H is bounded by a planeP. If P has non-empty intersection with the inside boundary of K(W), then P iscalled an inside bounding plane for W. The outside bounding planes are definedsimilarly. Notice that if P is an inside bounding plane, then OP intersects W inat least two points.

D.3. The inside, or outside, boundary of K = K(W) is a union of subsets ofboundary planes. Two planes intersect in a line, so in general there are lines ofintersection of boundary planes on OK. For general sets, three planes intersectin a point; however, this does not happen on 3K.

Proposition. Let W be a simple closed curve in §2, and let K = K(W). Let P, andP2 be inside boundary planes of K, where L =P, fl P2 # 0. Then L c OK.

Proof. Assume not. Then there is a third boundary plane P3, where P3 intersectsL at exactly one point. Normalize so that this point of intersection is at the originin 033. The three planes, regarded as Euclidean planes, divide §2 into eightspherical triangles. One of these triangles, call it T, contains W in its closure. LetC be the circle on §2 passing through the vertices of T, and let S be the hyperbolicplane whose boundary is C. Since the spherical length of any side of T is less


than n, S separates T from the origin. The half space bounded by S andcontaining T on its boundary contains K. This contradicts the assumption thatOeOK.

DA One can understand the above as saying that aK consists of some numberof faces, including degenerate ones (a degenerate face is just a line). Each face isthe intersection of aK with a boundary plane, and is a convex hyperbolic polygonwithout vertices (see IV.F); this includes the possibility of a degenerate polygon;i.e., a line. Of course, there might be sequences of boundary faces accumulatingto a boundary line of some other face.

A corollary of the above is that for every point x on OK, there is a completeline containing x, and contained in OK.

D.S. Proposition. Let L be a line in M. Both endpoints of L on l~2 lie on W.

Proof. Let x be an endpoint of L; suppose that x is not on W. Then there isa circular neighborhood U of x which also does not meet W. Let P be the planewhose boundary at infinity is W. One of the half spaces bounded by P containsW on its boundary; hence it contains K. We conclude that some points of L arenot on OK.

D.6. Proposition. Let G be a Kleinian group, and let W be a simple closed curve inQ(G), with closed inside disc B1, and closed outside disc B2. Suppose that Al isprecisely invariant under the subgroup H in G. Then the inside boundary ofK = K(W) is precisely invariant under H in G.

Proof. Let g e G. If g e H, then g preserves B1, so it preserves B2; from which itfollows that g preserves K(B2); hence it preserves the inside boundary of K(W).

If g#H, set W' = g(W), and call g(B1) the inside of W'. Then g maps theinside boundary of K onto the inside boundary of K' = g(K). A boundary planefor the inside boundary of K is a plane whose boundary at infinity contains B2;that is, the circle at infinity of the boundary of this plane is contained in B1.Similarly, a boundary plane for K' has, as its boundary at infinity, a circlecontained in g(B1) ). Since the inside of W and the inside of W' have disjointinteriors, these two boundary planes can intersect in at most one point on thesphere at infinity, (this is obvious in H', normalized so that oo lies outside bothW and W').

W.E. Discrete Groups of Isometries

E.I. A natural topology on L" is the compact-open topology; that is, the topologyof uniform convergence on compact subsets of H", or on t"-1. L" can also be

IV.E. Discrete Groups of Isometrics 67

regarded as a matrix group. L', the orientation preserving half of L2, is canoni-cally isomorphic to PSL(2;li); L3+ is canonically isomorphic to PSL(2;C); alsofor every n, L"+ is canonically isomorphic to a subgroup of index 2 in SO(n, I),the group of (n + 1) x (n + 1) matrices, with real entries and determinant 1,which keep invariant the form x; + + X.2 - xn+t .

The different views of L", as acting on H", or on 1'"-1, or as a matrix group,yield equivalent topologies on L". For n = 2 or 3, the equivalence of the matrixtopology of PSL with the compact-open topology on P-' is almost immediate.Since we will not use SO(n, 1), we will not prove the equivalence of this topologywith the others.

E.2. Proposition. Let { g,.) be a sequence of elements of V. g," -+ g uniformly oncompact subsets of H" if and only if gm -+ g uniformly on compact subsets of E"-'

Proof. It suffices to assume that g = 1. Let K be the closed ball of (hyperbolic)radius p about 0, and let z be some point of S"-'. To find gm(z), draw a hyperbolicline L from z through 0, and look at gm(L). Since gm(L) is (hyperbolically)uniformly close to L in K, it is (Euclideanly) uniformly close to L throughout G3";hence its endpoints are close to L on all of S"'.

Similarly, if gm(z) converges to the identity uniformly on compact subsets ofS"-', then gm(z) converges to the identity uniformly on S"-'. Locate a point xof a compact set K c H" by drawing two orthogonal hyperbolic lines, L and Mthrough x; then the point of intersection of gm(L) and gm(M) is g,"(x). Theendpoints of g,(L) and and gm(M) are uniformly close to the endpoints of L andM, respectively; hence gm(L) and gm(M) are (Euclideanly) uniformly close to Land M, respectively. Then, since K is compact, their point of intersection isuniformly close to x. 0

E3. Let X be one of the spaces H" (or, equivalently, a"), S" (or, equivalently, t"),or L", and let G be the group of isometrics of X.

Theorem. Let x be a point of X, and let G be a subgroup of G. G acts discontinuouslyat x if and only if G is a discrete subgroup of G.

Proof. If G is not discrete, then there is a sequence of distinct elements {gm} ofG, with gm - g e G. Then gm o g-' (x) -' x, so G does not act discontinuously at x.

Now assume that G does not act discontinuously at x. Let U be the ball ofradius p about x, and let gm be a sequence of distinct elements of G, withgm(U)fl u # 0 for every m. Then d(gm(x),x) < 2p; hence there is a subsequencewith gm(x) -* y e X. Find an element f e G with f(y) = x. Then f o gm(x) - x.Normalize so that x = 0. Since the orthogonal group is compact, there is asubsequence with D(f o gm)(x) -+ g e ®". It follows that f o gm - g. Hence G is notdiscrete. O


E.4. Corollary. G acts discontinuously at some point of X if and only if G actsdiscontinuously at every point of X.

E.5. Note that the proof in E.3 is independent of the radius p.

Proposition. If G is a discrete group of isometries of X, then for every x e X, andfor every p > 0, the ball of radius p about x contains only finitely many translatesof X.

E.6. As in II.E, there are two equivalent definitions of discontinuity.

Proposition. Let G be a discrete subgroup of G, and let x e X. Then StabG(x) isfinite, and there is a number p > 0 so that the ball of radius p about x is preciselyinvariant under Stab0(x).

Proof. For p' sufficiently small, the ball of radius p' about x contains no translateof x other than x itself. Set p = p'12; then the ball of radius p is precisely invariantunder Stab(x). Also observe that by E.3, Stab(x) is finite.

E.7. Another corollary of E.3 is that the set of points of X fixed by some non-trivialelement of G is nowhere dense in X. First, by C.2, the set of fixed points of eachg e G is a k-plane, with k < n. If the set of these fixed points were dense in someneighborhood U of x, then choose a neighborhood U' e U so that U' is preciselyinvariant under Stab(x). If the non-trivial element g e G fixes some point of U',then there are other points of U' that are not fixed by g; hence g e Stab(x). Thenumber of g in Stab(x) is finite, so the fixed points lie on at most a finite set ofhyperplanes through x.

W.F. Fundamental Polyhedrons

F.I. We continue the notation of the preceding section: X is one of the spacesH" (or B"), S" (or E"), or E", and G is its group of isometrics.

A hyperplane in X divides it into two half-spaces.A (convex) polyhedron D in X is the intersection of countably many open

half-spaces, where only finitely many of the hyperplanes, defining these half-spaces, meet any compact subset of X. The closure b of D has a natural celldecomposition given by the intersections of the defining hyperplanes. The k-cellsin this decomposition are called the k-faces of D, or of D. Also, the codimensionone faces are called sides, and the codimension two faces are called edges. Eachedge lies in the intersection of exactly two sides.

A polyhedron in dimension two is called a polygon; in this case, the co-dimension two faces are usually called vertices.

W.F. Fundamental Polyhedrons 69

F.2. Let G be a discrete subgroup of G. A polyhedron D is a fundamental poly-hedron for G if the following hold.

(i) For every non-trivial g E G, g(D) n D = 0.(ii) For every x c- X, there is a g e G, with g(x) a D.(iii) The sides of D are paired by elements of G; that is, for every side s

there is a side s', and there is an element g, a G, with g,(s) = s'. These satisfythe conditions: g, = g,-', and (s')' = s. The element g, is called a side pairingtransformation.

(iv) Any compact set meets only finitely many G-translates of D.

F3. Another way of expressing condition (iv) is to say that the tesselation of Xby translates of D is locally finite. If condition (ii) holds, then in order to checkif (iv) holds, it suffices to prove that every point of D has a neighborhood meetingonly finitely many translates of D.

F.4. If D is a fundamental polyhedron, then the identifications of the sides inducean equivalence relation on b; that is, x - y if there is a side pairing trans-formation g with g(x) = y. Condition (i) then says that no two points in D areequivalent. A consequence of condition (iv) is that each point of b is equivalentto at most finitely many other points of D.

F5. This equivalence relation also defines an equivalence relation on the edgesof D; each equivalence class of edges can be cyclically ordered as follows.

Start with an edge e,. It lies on the boundary of two sides, call one of themsi. Then there is a side si, and there is a side pairing transformation g withgi (s,) = s,. Set e2 = g, (e, ). Like e, , e2 lies on the boundary of exactly two sides,one of them is si, call the other s2. Again, there is a side s2, and a side pairingtransformation g2, with 9202) = s2. Continuing in this manner, we generate asequence {em} of edges, a sequence {g,,,} of side pairing transformations, anda sequence of pairs of sides. Since each point of e, is equivalent to atmost finitely many other points of D, the sequence of edges is periodic; hence allthree sequences are periodic. Let k be the least period so that all three sequencesare periodic with period k (except in the case that X = S", and D has exactly twosides, k is the least period of the sequence of pairs of sides).

The cyclically ordered sequence of edges {e1,. .. , ek } is called a cycle of edges;k is the period of the cycle. Two of these cycles are equivalent if they both containthe same set of edges; then one cycle can be obtained from the other by cyclicpermutation and/or by reversing the order in which the edges appear in the cycle.Of course, an edge can appear twice in a cycle, but not more than twice, for eachedge lies on the boundary of exactly two sides.

Each edge lies in exactly one equivalence class of cycles.Observe that gk o o g, (e,) = e, ; h = gk o o g i is called the cycle trans-

formation at e,.Let x be a relative interior point of e and let L be the 2-plane through


x orthogonal to e,. We look at the trace of the translates of D in L. Onthe other side of e, , we see g i' (D), then continuing around e,, we next see9i' o92 ' ogi(9i'(D)) = 9i 1 ogz(D), then gi o92' ogs'(D), and so on.

However, g o** o g, `(D) = h-` (D) might not equal D. The next translate wesee, continuing around e, is h ogI'(D), and so on. Eventually we come aroundback to D. We defined the period k so that when we do come back to D, thecorresponding translate of D, obtained from following the translates around, isa power of h-`. That is, there is a least positive integer q so that h9(D) = D. Thenh9 = 1.

The cycle transformation at an edge equivalent to e, is a conjugate of h, orof h-'; in any case, the order q depends only on the equivalence class of the cycle.

The two sides of D which meet at an edge e meet at a well defined angle a(e)measured from inside D. We have shown that if {e,,...,ek} is a cycle of edges,and the corresponding cycle transformation has order q, then

Y a(e.) = 2n/q.

F.6. Proposition. The side pairing transformations generate G.

Proof. If x is any point in X, then we can draw a path from some point 0 in Dto x, where the path does not pass through any translate of a codimension sface of D, s > 1. Then there is a conjugate of a side pairing transformationwhich maps each translate of D along this path to the next translate of D; theconjugating element is a product of side pairing transformations. 0

F.7. The space X has a Riemannian metric on it in which G acts as a group ofisometries; we can project this infinitesimal metric to Z = X/G, even though Zneed not be a manifold.

We are primarily interested in the distance function d(z,z') on Z, rather thanthe infinitesimal metric. This distance is defined to be the infimum of the lengthsof paths connecting z to z'. Equivalently, we can use the natural projectionp: X - Z, and define d(z,z') = infd(x,x'), where p(x) = z, and p(x') = z'.

We can reconstruct this distance function in D as follows. Let x and x' bepoints of D, where p(x) = z, and p(x') = z'. Then d(z, z') = inf E d(xm, x,,), wherethe infimum is taken over all finite sets of points {x xi,...,xj,xj} in D, withp(x,) = z, p(x;,,) = p(x.+1), and p(xj) = z'.

It is easy to see that with this definition of distance, the space Z = %/G is acomplete metric space.

W.G. The Dirichlet and Ford Regions

G.I. We continue with X and G as above. Assume that G is a discrete subgroupof G, and choose a point 0 in X which is not fixed by any non-trivial element of G.

W.G. The Dirichlet and Ford Regions 71

For any non-trivial element g e G, the perpendicular bisector of the linejoining 0 to g(O) is a hyperplane. Let D9 be the half space of points closer to 0than to g(0); i.e., D9 = {x e X I d(x, 0) < d(x, g(0)) }. The Dirichlet region D, centeredat 0, is the intersection of all the half spaces D9.

Since there are only finitely many points of the form g(0) in any compact set,D is a convex polyhedron. Also observe that since g(0) = 0 only for g = 1, eachside of D corresponds to a unique element of G.

G.2. Theorem. The Dirichlet region D is a fundamental polyhedron for G.

Proof. If g is a non-trivial element of G, and x is some point of D, then d(g(x), g(0)) =d(x, 0) < d(x, g'(0)) = d(g(x), 0); hence g(x) is not in D. This in condition (i)of F.2.

To prove condition (ii), let x be some point of X. Then there is a (not neces-sarily unique) element g with d(x, g(0)) < d(x, h(0)) for all h e G. Then, writingan arbitrary element of G as g o h, d(g-' (x), 0) = d(x, g(0)) 5 d(x, g o h(0)) =d(g-' (x), h(0)) for all h e G. It follows that g-`(x) lies in D,, for every h; hence itlies in D.

Let x be a point of the relative interior of a sides of D. Then there is a uniqueg e G with x e b,; i.e., d(x, 0) < d(x, h(0)) for all h # g, and d(x, 0) = d(x, g(0)). Thend(g-' (x), 0) = d(x, 0) = d(g-' (x), g-' (0)), and for any h :A g-', d(g-' (x), h(0)) =d(x, g o h(0)) > d(x, 0) = d(g-`(x), 0). Hence g-'(x) also lies on a side s' of D. Itfollows that g-1(s) = s'.

Let K be compact; we can assume that K is the closed ball of radius p about0. It follows from E.5 that there are only finitely many translates of 0 within theball of radius 2p. If d(g'(0), 0) > 2p, then g(D) n K = 0. 0G3. Let G be a discrete subgroup of L"+', viewed as acting on Gz", where oQ isnot fixed by any element of G. For each g e G, consider the family of Euclideanhyperspheres centered at g-'(oo). Reflection in one of these spheres C inter-changes oo and g-'(oo). Hence reflection in g(C) interchanges g(oo) and oo; sog(C) is a Euclidean sphere centered at g(oo). It is clear that the radius of g(C) isa continuous and monotone decreasing function of the radius of C. Hence thereis a unique hypersphere 19 in this family, so that 19 and g(19) have the same radius.19 is called the isometric sphere of g.

It is clear that the isometric sphere is unique. Hence g(19) = 19 = 19Assume first that 19 # 19. Denote reflection in 19 by p, and let q be the reflection

in the Euclidean perpendicular bisector of the line joining g-'(oo) to g(oo).Observe that r = g o p o q has fixed points at g(oo) and oo; it also preserves thehypersphere 19. We conclude that r is a Euclidean rotation centered at g(oo); i.e.,as an element of L"', r is elliptic, and the hyperbolic line in H"I with endpointsat g(oo) and oo is kept pointwise fixed by r.

Assume next that to = 19. Set r = go p, where p is reflection in 19. Exactly asabove, observe that r is a Euclidean rotation centered at g`(oo) = g(oo).

72 IV. Groups of Isometries

We have shown that g = r o q o p, where p is reflection in I. and r and q areEuclidean motions.

For every point x, Dg(x) is a positive multiple of an orthogonal matrix. Wewrite that multiple as IDg(x)l.

The isometric sphere 1e divides I" into two discs; the outside which containsoo, and the inside. Since g can be written as the composition of p and a Euclideanmotion, I. is the set of points for which the differential Dg satisfies JDgl = 1; theoutside of Ia is the set of points where IDgI < 1, and inside I., IDgl > 1.

For each g E G, let D9 be the outside of I, and let D be the interior of theintersection of the closures of all the D9, g # 1. D is the Ford region for G.

G.4. As in II.H, we could prove that the Ford region is a fundamental domainfor G (a concept thus far defined only in dimension two). However, we do thisonly for discrete subgroups of P".

G.5. We can write an arbitrary element g E P" as g = h o r, where h is hyperbolicand r e O. To see this, draw the hyperbolic line segment L from 0 to g(0), andlet P be the perpendicular bisector of L. Let Q be the hyperplane through 0 whichis orthogonal to L. Define h-' as the composition of reflection in P followed byreflection in Q. Since P and Q are both perpendicular to L, they cannot intersect(to see this, normalize so that L is the line in H" from 0 to oo ). Thus his hyperbolic.Since h-'og(0)=0,r=h-'oge®".

G.6. Proposition. If G is a discrete subgroup of P", where Stab(0) is trivial theninside B", the Ford region and the Dirichlet region centered at 0 coincide.

Proof. It suffices to show that for every g e G, P9, the perpendicular bisector ofthe line joining 0 to g-'(0), and 1B coincide (inside B", of course).

Write g"' = r-' o h-', where r E 0", and h is the composition of the reflectionin Pa, followed by the reflection in a Euclidean hyperplane. Then IDg-'(x)I _(Dh-'(x)I = I precisely on Pa.

G.7. Proposition. Let G be a discrete subgroup of P", where Stab(0) is trivial. Letp be the Euclidean radius of the isometric sphere of g e G. Then

z,p2n<oo.

Proof. Let U be a spherical disc about oo of the form (x e X I I x I > a) U {00 ),where U is precisely invariant under the identity in G. Then U g(U) is the set ofall translates of U under G, and these sets are mutually disjoint. Since they aredisjoint, we can exclude U and sum the Euclidean volumes of the others; i.e.,Y'(dia,(g(U))" < oo.

Write g = sop, where s is a Euclidean motion, and p is reflection in theisometric sphere I of g. Let p be the radius of 1.

W.H. Poincare's Polyhedron Theorem 73

Observe that the proof of I.C.7 is independent of dimension; that is, if p is theradius of 1, 6 is the distance from the center of I to aU, where U contains oo, andp denotes reflection in 1, then

pz/S 5 diaE(p(U)) 5 2pz/b.

Apply this to the reflections in the isometric spheres of the elements ofG - (1), to obtain

00 > (diaE(g(U))" p2./,j. > F, pzn/&' = 6-" pzn.

G.B. Corollary. Let {g",} be a sequence of distinct elements of the discrete subgroupG c P", and let I,,, be the isometric sphere of g,". Then dia(Im) -+ 0.

G.9. Proposition. Let G be a discrete subgroup of P". Let {g",} be a sequence ofdistinct elements of G, with gn,(0) -, x e Sn-1. Then there is a subsequence, anda point y e Sn-1, so that gm(z) -' x uniformly on compact subsets of C" - { y}.

Proof. As in II.D.2, choose the subsequence so that the center of the isometricsphere I. converges to some point, call it y, and so that the center of the isometricsphere 1 also converges, necessarily to x. The result now follows from the factthat g", maps the outside of 1,, onto the inside of I,", together with the fact thatthe radii of these spheres tend to zero.

W.H. Poincare's Polyhedron Theorem

H.I. We continue our assumptions on X and G; that is, X is one of the spacesH" (or 0"), E", or Sn (or 2"), and G is the group of isometrics of X. We also assumethat n ;?: 2.

Assume that we are given a polyhedron D, where the sides of D are pairwiseidentified by elements of G; our goal is to write down conditions on D toguarantee that the group G, generated by the identifications of the sides of D, isdiscrete, and that D is a fundamental polyhedron for G.

H.2. The first condition is that the sides of D are paired by elements of G. Thatis, we assume that for each side s of D, there is a side s', not necessarily distinctfrom s, and there is an element g, a G, satisfying the following conditions.

(i) g:(s) = s'.(u)93.=gs1.The isometrics g, are called the side pairing transformations.Since s and s' are both sides of D, g,(D) and D either both lie on the same side

of s', or they lie on opposite sides. If they lie on the same side, then of course,g,(D) n D # 0; this gives us our third condition:

(iii) g,(D) fl D = 0.


H3. Let G be the group generated by the side pairing transformations. Observethat if there is a side s, with s' = s, then condition (ii) implies that g, = 1. If thisoccurs, the relation gs = 1, is called a reflection relation.

H.4. The side pairing transformations induce an equivalence relation on D, whereeach point of D is equivalent only to itself. Let D* be the space of equivalenceclasses, with the usual topology, so that the projection p: D -+ D* is continuousand open. If D is to be a fundamental polyhedron for G, then condition (iv) ofF.2, requires that there be only finitely many points in each equivalence class ofpoints of D.

(iv) For every point z e D*, p-1(z) is a finite set.

H.5. Our next two conditions are related to the edges. Exactly as in F.5, the edgescome in cycles; the condition above guarantees that each cycle is finite. For eachedge e = e,, let {e1, ... , ek } be the ordered set of edges in the cycle containing e(as in F.5, k is chosen to be the least period so that the sequences of edges, pairsof sides, and side pairing transformations all have period k), and let gl,..., gk bethe corresponding side pairing transformations. Then the cycle transformationh = h(e) = gk o o g, keeps e invariant. As in F.5, h depends on a choice of a sideabutting e; if we choose the other side to start with, then we obtain h-` as thecycle transformation.

(v) For each edge e, there is a positive integer t so that h' = 1.The relations in G, of the form h' = 1, are called the cycle relations. There is

essentially only one cycle relation for each equivalence class of cycles. If e' isequivalent to e, then h(e') is a conjugate of (h(e))I'.

Continuing as in F.5, we let a(e) be the angle, measured from inside D, at theedge e. We require

k

(vi) E a(em) = 2n/t.ni=,

H.6. The conditions listed so far are sufficient to guarantee that if we look onlyat D, and those translates of D that we know to abut D, then the closures ofthese fit together without overlap, except along the translates of the sides, to fillout a neighborhood of D; this is the content of H.12.

H.7. In order to state the last condition, we need the following construction.We first form the group G*, defined to be the abstract group generated by

the side pairing transformations, and satisfying the reflection and cycle relations;we also endow G* with the discrete topology. There is an obvious homomor-phism c: G* -+ G.

We next consider the equivalence relation on G* x D generated by thefollowing. The pairs (g;, x,) and (g2, x2) are equivalent if there is a side pairingtransformation f with f(x1) = x2, and if, as elements of G*, g2 = g; of -1. LetX* be G* x D, factored by this equivalence relation. We endow X* with the

W.H. Poincare's Polyhedron Theorem 75

usual identification topology, so that the natural projection from G* x D to Xis continuous.

We remark that it is not at this point clear that this equivalence relation islocally finite. That is, there might be infinitely many points of the form (g.*, x) inG* x D which are all identified in X*.

H.8. There is a natural map q: X* -. D*, defined by projection on the secondfactor of G* x D, followed by the projection p from b to D*. It is easy to see thatq is well defined and continuous.

There is also a map r: X* - X, defined by r(g*,x) = a(g*) (x). It is easy tosee that r is well defined and continuous; our eventual goal is to prove that r isa homeomorphism, and incidentally, that a is an isomorphism.

One should view X* as the set of translates of D under G*, where thesedifferent translates have been sewn together at the sides so that the map r is welldefined. We should also think of X* as the set of translates of b under the groupG, where we regard overlapping, other than that given by the identifications ofthe sides, and the known relations of G (i.e., the relations of G*), as lying ondifferent sheets over X; then r is the projection from this covering to X. (Thinkof a side pairing transformation g as defining "analytic continuation" from b tog(D); then X* is the "Riemann surface" of the "function" defined by this analyticcontinuation.)

H.9. In the lemma below, we prove that r is a local homeomorphism. Once wehave established this, we can use r to lift the local differential metric from X toX*; then the distance between points of X* is the infimum of the lengths ofsmooth paths joining them. We use this distance on X* and the projection q todefine a distance on D*; the distance d(z, z') between points of D* is the infimumof the distances d(x, x'), where q(x) = z, and q(x') = z'.

It is easy to see that this is the natural notion of distance on D*; that is,d(z, z') = inf Z d(xm, x,,), where the infimum is taken over all finite sets of points

(x1,xj,..., Xk,Xk), in D, with p(x1) = z, P(x,) = P(Xm+t ), and p(xk) = z' (see F.7).

H.10. Our last condition is(vii) D* is complete.

H.1 1. Theorem. Let D be a polyhedron with side pairing transformations satisfyingconditions (i) through (vii). Then G, the group generated by the side pairing trans-formations is discrete, D is a fundamental polyhedron for G, and the reflectionrelations and cycle relations form a complete set of relations for G.

H.12. Lemma. Let D be a polyhedron with side pairing transformations satisfyingconditions (i) through (vi). Then every point z* e D* has a neighborhood U so thatq-1(U) is a disjoint union of relatively compact open sets U., where for each a, rl U.is a homeomorphism onto a convex set.


Proof. Notice first that G* acts as a group of homeomorphisms on X*, and that1 x D is a "fundamental domain" for this action; that is, no non-trivial translateunder G* of 1 x D intersects it, and the union of the translates of the closurecovers all of X*. We also remark that this lemma asserts that the translates ofD, under those side pairing transformations that are known to abut D, preciselyfill out a neighborhood of D.

If x is an interior point of D, then let b be the distance from x to the nearestside, and let V be the ball of radius b about x. Set U = p(V). Since every pointof V is equivalent only to itself, the preimages of V in X* are precisely the setsof the form U,, = g* x V; these are disjoint open sets, and for each a, rl U. isa homeomorphism onto a ball of radius S.

H.13. If x is an interior point of a side s of D, then there is another side s', andthere is a side pairing transformation g with g(s) = s'; set x' = g(x). If x x', letb be the minimum of the distance from x to x', the distance from x to any sideof D other than s, and the distance from x' to any side of D other than s'. Let V(V') be the intersection of the ball of radius 6/2 about x (x'), with D. Note thatV and V' are disjoint. Set U = p(V) U p(V').

If x = x', let b be the minimum distance from x to any side of D other thans, let V be the intersection of the ball of radius 6/2 about x with b, and letU = p(V).

Each connected component U. of q-1(U) consists of the union of two halfballs. If x jA x', then near (1, x), these are the half balls I x V and g-' x V'. Ifx = x', these are 1 x V and g' x V. Since x' is the only other point of Dequivalent to x, each U. is a neighborhood of a point of the form (g*,x) in X*;it is clear that ri U. is a homeomorphism onto a ball of radius 6/2.

H.14. Next let x = x1 be an interior point of an edge et. Let {e1,...,ek} be thecycle of edges containing e1, let h = g,t o o g1 be the cycle transformation at e1,and let t be the order of h. Define the tk elements of G*, j1, ...,ilk, byj1 = 91,J2 = 92 091, ..., Jk = h, jk+1 = 91 oh, ..., jtk-1 = 9k-1 o... og1 oh", ilk = 1. Letxm+1 = jm(x1). Each of the points x, lies in the intersection of two sides; let 8,,, bethe minimum of the distance from x,,, to any other side of D, and of the distancefrom xto any point xi # Let 6 = 1/2 and let V. be the intersectionof the ball of radius S about x, with D; observe that the sets V. are all disjoint.Set U = U p(Vm). Each component U,, of q-' (U) is a union of tk "wedges". Near(1,x), these wedges are the sets (1 x V1), ..., (jrk'-1 x Vk).

Each edge lies in the intersection of exactly two sides, and each side uniquelydetermines its side pairing transformation. It follows from condition (v) that

1,xtk)} is a complete set of equivalent points of G* x D.Condition (v) also implies that the set of the form U. near (1, x) is a neighborhoodof (1, x). Condition (vi) asserts that, in the 2-plane orthogonal to e1, these tktranslates of D fit together without overlap, and fill out a neighborhood of x inthat plane. It follows that rJU, is a homeomorphism onto a ball of radius 6.

W.H. Poincare s Polyhedron Theorem 77

Using the action of G* on X*, we see that the same statement is true for anarbitrary point of the form (g*, x).

H.15. Next suppose that for every p, 2 5 p < j3, and for every point x, which isan interior point of a codimension p face of D, there is a neighborhood U of p(x),so that our lemma holds. Let F be a codimension ¢ face, and let x be an interiorpoint of F. Let x = x,, X2,..., x,, be the points of b that are equivalent to x, andlet j be an element of G* with x, (we use the same notation for elementsof G and for elements of G*; this should cause no confusion). Each xm lies in theintersection of some number of sides of D; let b. be the minimum of the distancefrom x,, to any other side of D, and of the distance from x, to any x&, k # m. Letb = 1/2 min(b,.). Let P be the codimension P face on which xm lies, and let F.be the ft plane orthogonal to F. at xm. Let 17, be the intersection of the ball ofradius b about xm with F. and for each a < b, let IV. be the intersection of F.with the ball of radius b about xm,. Let 0 = U p(P,), and let 0 = U p(19.). Then0 is the restriction of a neighborhood of z = p(x) to p(F,) = . . . = p(Fm). Thereis no difficulty showing that the components of q-'(0) are relatively compact,and that r, restricted to any one of them, is a homeomorphism.

The boundary of p-'(0) lies entirely in codimension p faces of D, p < p,and in D. Hence for every component 0, of q-'(0), rIa0, is not only a localhomeomorphism, but a covering of its image. Near the point (1, x), the image ofrIa0, is easily seen to be the entire ft-sphere of radius S about x. Since ft z 2,rIa0, is a homeomorphism.

The argument above shows that there are no translates of I x x on I x D inX*, other than the obvious ones. It follows that each component 0, is relativelycompact in X*.

The argument above is independent of the radius 3, hence, for each x, rl 0, isa homeomorphism. Set U = 0 x 0, and observe that for each a, U. is relativelycompact in X*, and rI U. is a homeomorphism onto a product of discs, which isconvex. 0

H.16. Corollary. Let D be a polyhedron with identifications satisfying conditions(i) through (vi). Then r: X* -+ r(X*) is a covering.

H.17. Our goal is to show that r is a homeomorphism, in fact an isometry. For,if we show this, then we will certainly have that a is an isomorphism; nonon-trivial translate of D intersects D; and the union of the translates of D coversX. We assumed to start with that the sides of D are pairwise identified by elementsof G. Finally, we showed in H.15 that only finitely many translates of 1 x Din X* intersect it at any point, from which it will follow that the tesselationof X* by translates of I x D is locally finite. Once we have proved that r is ahomeomorphism, this is equivalent to the statement that the tesselation of X bytranslates of D is locally finite.


1-1.18. Let b be some point in the image of r, and let w be a geodesic emanatingfrom b. Since r is a local homeomorphism, we can locally lift w to X* near b.Parametrize w by arc length s and assume there is some first point so, so that fors < so, we can lift w(s) to a path '(s) on X*, but we cannot continuously lift w(s),0 5 s 5 so, to X*. Choose a sequence {sm) of numbers, where sm -+ so, so thatfor every m, the segment of w(s) with sm 5 s 5 sm+, lies in an open set of the formU., as in H.12.

Using the convexity of r(UQ), it is easy to see that if x and x' are any twopoints of U then d(q(x), q(x')) 5 d(r(x), r(x')). Hence

Y d(q o 0(s.), q o w(s.+i )) 5 Y d(w(sm), w(sm+i )) < oo.

Since D* is complete, the points q o w(sm) converge to some point z; in fact,q o w(s) -+ z. For s sufficiently close to so, q o w(s) lies in a neighborhood U of zas in H.12-15. Hence w(s), for s sufficiently close to so, lies in one of the relativelycompact sets of the form U. Hence we can continuously define the lift w(so). Wehave shown that we can lift the entire geodesic w.

Except on a sphere, the lifting of geodesics defines a global inverse to the localhomeomorphism r; hence r is a homeomorphism. On a sphere, r' is well definedin any ball of radius less than it about b; the result again follows.

H.19. Corollary. Let D be a fundamental polyhedron for the discrete group G. Thenthe identifications of the sides of D generate G, and the reflection and cycle relationsform a complete set of relations for G.

H.20. We remark in conclusion that the convexity of D was used only in thedefinition of a polyhedron. One can prove the same theorem for polyhedra thatare not necessarily convex; the main difficulty lies in defining such an object.

W.I. Special Cases

I.I. There are several special circumstances under which one or more of con-ditions (i)-(vii) of the preceding section are automatically fulfilled. In this section,we describe some of these special circumstances.

Conditions (i)-(iii) are basic; we assume throughout this section that we aregiven a polyhedron with identifications satisfying these three conditions.

1.2. In dimension 2, each edge is just a point, so condition (iv) is automaticallysatisfied if each cycle of edges is finite. In particular, condition (iv) is satisfied, indimension 2, if D has finitely many sides.

1.3. Continuing with the case that X has dimension 2, observe that condition (v)is a consequence of (vi). That is, condition (vi) is used to show that each cycle

IV.I. Special Cases 79

transformation, when raised to an appropriate power, is the identity in the2-plane orthogonal to the edge. In dimension 2, that is equivalent to showingthat this power of the cycle transformation is the identity.

1.4. Proposition. Let X have dimension 3, and let D c X be a relatively compactpolyhedron with orientation preserving side pairing transformations, where all con-ditions of IV.H, other than (v) are satisfied. Then (v) is also satisfied.

Proof. Let h = 91 o o gk be a cycle transformation at the edge e; assume thatthe sum of the angles of the k edges in the cycle containing e is 2n/t. In the 2-planeorthogonal to e, h' is the identity; in particular h` preserves orientation in that2-plane. Since h is orientation preserving, h' preserves orientation on e. Finally,since e has finite length, h' I e = 1.

I.S. Proposition. If D is relatively compact in X, and D, with side pairing trans-formations, satisfies conditions (i) through (vi) of IV.H, then (vii) is also satisfied.

Proof. Since D is relatively compact in X, D* is compact; hence complete.

1.6. For finite sided polyhedra, one can restate the completeness condition interms of the identifications of those sides that extend to infinity. For the re-mainder of this section, we assume that D has only finitely many sides; we alsoassume that conditions (i) through (vi) are satisfied.

As we have already remarked, there is nothing to prove if X is a sphere. It iseasy to see that every polyhedron in l_" satisfying conditions (i)-(vi) also satisfiesthe completeness condition, (vii). It is only in H" that one can have two sides withzero distance between them, even though they do not intersect. We now assumethat X = H".

We might have two sides that are tangent at some point x = x1 on the sphereat infinity; call one of these sides s,. Let g, be the side pairing transformationwith g,(sl) = s;, and let x2 = g1(x0. If x2 is not also a point of tangency of s;and some other side, there is nothing further to do; if it is, then call the other sides2, find the side pairing transformation 92 with g2(s2) = S2, set X3 = 92(X2), andcontinue. If, after a finite number of steps, we return to x = x, (that is, we findside pairing transformations g1, ... , gk with h(x) = gk o O g&) = x) then wecall x an infinite edge, and we call h the infinte cycle transformation at x.

Proposition. Assume that the finite sided polyhedron D c H" satisfies (i)-(vi) ofIV.H. Condition (vii) is also satisfied if and only if every infinite cycle trans-formation at every infinite edge is parabolic.

Proof. We first assume that every infinite cycle transformation is parabolic.Assume that there is a sequence Ix.) of points of D, where z,, = p(x,") is a Cauchysequence, and x,, - x, a point on the sphere at infinity.


We can assume that all the points xm lie on some side s of D. Consider thegeodesic in X* (see IV.H) from (1, xm) to (gm, x",+1), where (gm, xm+,) is chosen sothat d(zm, zm+,) = d((l, xm), (gm, xm+, )).

It is easy to see that there is nothing to prove unless x either lies at the endof an edge, or is an infinite edge.

If x is an infinite edge, then normalize so that x = oo. Since D has only finitelymany sides, for m sufficiently large, the element gm is necessarily a power ofthe infinite cycle transformation h at x. Since h is parabolic, it is an element of!A"; i.e., h e A"+', and h keeps E" invariant. It follows that xm+, and gm(xm+,) havethe same height. We conclude that d(zm, zm+,) Z d(xm, xm+, ). Since {zm } is aCauchy sequence, {xm} converges in H".

If x is not an infinite edge, then we can assume that all the points xm lie onsome edge e. In this case, for m sufficiently large, the element gm is some powerof the cycle transformation at e. Since the cycle transformation has finite order,it is elliptic. Now normalize so that D lies in B", and so that 0 is a fixed pointof the cycle transformation stabilizing e. Then that cycle transformation is arotation in 0"; hence, using the same notation as above, xm+t and gm(xm+,) bothlie at the same (hyperbolic and Euclidean) distance from the origin. Then,d(zm, Zm+,) Ixm - x,,,+, I; since {zm } is a Cauchy sequence, {xm} is convergentin W.

1.7. We now assume that there is an infinite edge x, whose cycle transformationh is not parabolic. Let s, be one of the sides abutting x, and let s, be the other.Write h = ga o o g,, as in the definition. Then, exactly as with ordinary edges,the translates of D near x are D, then g '(D), then g;' o g2' (D), and so on, up toh-'(D). Hence, s, separates D from h(D), and h has infinite order. Thus h isloxodromic.

Normalize so that x = oo, and so that h has its second fixed point at 0; i.e.,h = rod, where r is a rotation, and d is a dilation. Replace h by its inverse, ifnecessary, so that d(y) = Ay, A > 1. Since s, passes through oo, it is vertical;hence there is a sequence of points {ym} = ((a, A")) on s,, where a is some fixedpoint of t", and m e Z. Note that the points h(ym) all lie on a vertical line. LetK be the Euclidean distance I a - r(a)I. Then d(h(ym), ym+,) is less than the integralof the hyperbolic metric along the Euclidean line between these points; i.e.,d(h(ym),Ym+,) < KA-(m+'). Since d(h(ym),ym+1) dominates the distance on D*between the projection of ym and the projection of ym+,, D* is not complete.

0

W.J. Exercises

J.1. Every circle C in C lies on the boundary of a plane Pin H'. Let g e IYO denotereflection in C. Then g, as an element of 1.', is the reflection in P.

IVJ. Exercises 81

J.2. What is the relation between A.9, for n = 3, and I.A.8?

J3. Let j denote reflection in "-'. Then j commutes with every element of V.

J.4. If C is a Euclidean k-sphere whose closure is contained in H", n > k, then Cis a hyperbolic k-sphere (a hyperbolic k-sphere is the intersection of a hyperbolicball with a hyperbolic k-plane passing through the center of the ball).

JS. For every k = 0, 1, ..., n - 2, there is an elliptic element of L" whose fixedpoint set is a k-sphere in t"-'.

J.6. (a) Every elliptic element of L3 can be written as a product of at most threereflections.

(b) There is an elliptic element of L3 which cannot be written as a product offewer than three reflections.

J.7. A parabolic element of L" can be written as a product of exactly two reflec-tions if and only if it is a translation.

J.S. Every element of L" of finite order is elliptic.

J.9. If g E M is parabolic, elliptic, hyperbolic or loxodromic, respectively, then g,as an element of L3 is respectively, parabolic, elliptic, hyperbolic or loxodromic.

J.10. Every hyperbolic element of P' can be written, as an element of NA, in theform

a2-b2=1.

J.11. Let x # y be two points on OH" = t"-'. There is a unique geodesic whoseendpoints, on the sphere at infinity, are x and y.

J.12. Let x e H", and let y e aH". There is a unique geodesic passing through x,and having one endpoint at y.

J.13. (a) Show that the (real) dimension of L"+' is l + 2n + n(n - 1)/2.(b) The elliptic elements of L"+' have codimension one.(c) The parabolic elements of L"+' have positive codimension; either 1 or

2, according as n is odd or even. (Hint: consider only orientation preservingtransformations.)

J.14. Consider the transformation g(z) = a + 1 + i, as a parabolic element of V.There is a parabolic element h e L' in standard form which is conjugate in L3 tog. Is there an orientation preserving element of L3 which conjugates g into h?


J.15. Let G be a discrete subgroup of A", where every element of G is a translation.Then G is an Abelian group of rank at most n.

J.16. Let G be a discrete subgroup of L3+, where G contains the transformationz -+ z + 1. Then {(x,t)It > 1} is precisely invariant under Stab(co) in G. (Hint:see II.C.5.)

J.17. A Euclidean ball in H"+' whose boundary is tangent to E" at x is calleda horoball at x. Use J. 16 to prove that if G is a discrete subgroup of L3' containinga parabolic element with fixed point x, then there is a horoball at x which isprecisely invariant under Stab(x).

J.18. By making appropriate changes in the statement and proof of II.C.5, provethe results of J.16, and J.17, with Ls+ replaced by V.

J.19. Let G be a discrete subgroup of P", and let {g",} be a sequence of distinctelements of G. For every x c-!5'-', the set of points z e B" for which g",(z) - x isboth open and closed in B".

J.20. Let C, , C; , ... , Ck, C be 2k disjoint (n - 2)-spheres in E"-1 with a commonexterior. For each m, let g", be an element of L" mapping C. onto C where g",maps the inside of C,, onto the outside of C,,. Use Poincare's theorem in H" toshow that G = <g1,... , gk> is discrete and free on these k generators (for n = 3, Gis a classical Schottky group of rank k).

J.21. Let C1, ..., C be as in J.20, except that for certain m we permit C. to betangent to C,,, and we require the corresponding g", to be parabolic (necessarilywith fixed point at the point of tangency). In this case as well, G is discrete andfree (for n = 3, such a group is called a Schottky type group).

J.22. The Picard group PSL(2, 1(i)) is the subgroup of u consisting of allunimodular matrices whose entries are Gaussian integers (i.e., complex numbersof the form a + ib, a, b c- 1). Let P be the polyhedron formed by the (hyperbolic)lines whose boundaries on the sphere at infinity are the following: {zIIzj = 1},{zIRe(z) = -1/2}, {zIRe(z) = 1/2}, {zIIm(z) = 1/2}, and {zIIm(z) = 0}.

(a) Use this polyhedron P to prove that the following elements generate thePicard group: z - - z, z - z + 1, z - -1/z, and z - - z + i (Hint: use isometriccircles to show that P in precisely invariant under the identity).

(b) Find a complete set of relations for PSL(2,1(i)).

J.23. Conditions (i), (ii), and (iv) of IV.H are basic in that the other conditionsneed not make sense without them. Show that the other conditions are allindependent.

W.K. Notes 83

J.24. Let G be a discrete subgroup of P". Then Y I 1 - g(0)1" < oo. (Hint:1 - g(0)I < Ja - g(0)I, where a is the center of the isometric sphere of g.)

W.K. Notes

A3. Further information about hyperbolic geometry can be found in the articleby Milnor [73], and the references listed there. D.3. The boundary of the convexset K(W), where W is the limit set of a quasifuchsian group of the first kind, hasbeen extensively studied by Thurston [90]. E.I. The different topologies on Iare discussed in Beardon [11 pg. 45-54]. Discrete groups of motions of then-sphere, and Euclidean n-space, are studied in Wolf [99]. F.2. The proof of H.11does not make essential use of convexity; non-convex fundamental polygons areused by Keen [39]. H.11. This theorem is due to Poincare in dimensions 2 [79]and 3 [80]. The proof here follows [63]; see also Morokuma [75]. Another lineof proof can be found in de Rham [21]. There is a generalization to higherdimensional complex spaces by Mostow [76]. J.17. It was independently ob-served by B.N. Apanasov and P. Waterman that this fact does not generalize tohigher dimensions when the parabolic element has an irrational twist.

Chapter V. The Geometric Basic Groups

The geometric basic groups are those Kleinian groups which are also discretegroups of isometries in one of the 2-dimensional geometries discussed in the lastchapter. That is, a geometric basic group is a (conjugate of a) discrete group ofisometries of S2, [2, or I-I2.

In this chapter we discuss the three cases, classify the elementary groups (theseare Kleinian groups with at most two limit points), and give some applicationsto non-elementary groups.

V.A. Basic Signatures

A.I. A component of a Kleinian group G is a connected component of the set ofdiscontinuity, Q(G). A component d of G is invariant if g(d) = A for all g e G. AKleinian group G with an invariant component A is a function group if d/G is afinite marked Riemann surface. The definition of function group depends on thechoice of invariant component (there might be more than one); we sometimeslabel the function group as (G, A).

A.2. Let (G, d) be a function group, and let S = d/G. Set °d = d n v, and setp(°A) °S. Then 'S can be conformally embedded in a compact Riemann sur-face S of genus p, where S - °S consists of a finite number of special points{x,__ x }. These points are either ramification points of the projection p: A - S,or they are the points of S - S. Each of these points has a ramification numberassociated to it. If xm is in the image of p, then the ramification number vm is theorder of the stabilizer of a point lying over xr,. If x, is not in the image of p, thenthe ramification number v,, = oo.

The basic signature of (G, A) is the collection of numbers (p, n; v...... v.), where2 < v, < oo. We usually write the basic signature so that v, < . < v,,.

A3. If (G, A) is a function group, and A is simply connected, then p: A -+ d/G isa branched universal covering (see III.F). If (0, 3) is another such group with thesame basic signature, then there is a homeomorphism tp: d/G - a/r';. Thishomeomorphism lifts to a homeomorphism q: A --*.3, where 0 o g o Q' defines

V.B. Half-Turns 85

an isomorphism of G onto C. The homeomorphism 0 is called a similarity, andwe say that Q induces the isomorphism: g - i o g o 0'.

A.4. As usual, a presentation <a, b.... :0 ° = vO = = I> of an abstract groupG is a set of generators, a, b, ... , and a set of relations, u° = 1, va = 1, ... , whereu, v, ... , are words in these generators, and a, fi, ... are positive integers. This setof relations is required to be complete; that is, if we look at the free group F onthe generators a, b, ... , and let N be the smallest normal subgroup of F contain-ing the elements ua, 0,..., then N is the kernel of the natural homomorphismfrom F onto G (the natural homomorphism sends the generator a of F to the gen-erator a of G, etc.). We will refer to this concept of-presentation as an abstractpresentation.

We remark that we also use the notation <a, b,... > to denote the groupgenerated by a, b, ... ; this should cause no confusion.

A.5. In some sense, the general element of a Kleinian group is loxodromic. If weknow a presentation for a Kleinian group G, then we know all elliptic elementsof G. We enlarge our concept of presentation to include "relations" which assertthat certain elements of G are parabolic.

A presentation of a Kleinian group G is a set of symbols as before: <a, b,... :u° = vB = - = I>, where again a, b, ..., are generators of G, and u, v, ..., arewords in these generators, but now we permit the exponent to take on the valueoo.

This presentation has the following meaning. If we look only at those rela-tions with finite exponent, these form an abstract presentation for G. The symbolu°° means that u is parabolic in G, and that the maximal parabolic subgroup ofG containing u has rank 1(we will see below that every purely parabolic Kleiniangroup is Abelian). Also if h is any parabolic element of G, where the maximalparabolic subgroup containing h has rank 1, then h is conjugate in G to a powerof an element u, where the symbol u' appears in the presentation.

A.6. The restriction in the above to parabolic elements which lie in rank Iparabolic subgroups is fairly natural. In a Kleinian group, every free Abeliansubgroup of rank two is purely parabolic. Hence the abstract presentationalready tells us which elements of a Kleinian group lie in a rank 2 purely parabolicsubgroup.

V.B. Half-Turns

B.1. An elliptic element of M of order 2 is called a half-turn. If a e IDA has exactlytwo fixed points on C, then the hyperbolic line in H3 joining these points is theaxis of a. If a is elliptic, then every point on the axis Aa is fixed by a; we also saythat a is the half-turn about Aa.

86 V. The Geometric Basic Groups

B.2. Proposition. Let g be an element of MI, where g is not parabolic, and let A be ahyperbolic line in 0-03 orthogonal to Aa. Then g = boa, where a is the half-turn aboutA, and b is a half-turn about a line B, orthogonal to Ao. Further, Ag, A and Bintersect at a point in H3 if and only if g is elliptic.

Proof. The axis of g is of course invariant under g, and the endpoints are the fixedpoints of g. Since A is orthogonal to A9, a o g preserves Ae and interchanges itsendpoints; hence a o g = b is a half-turn. A half-turn preserves the hyperbolic lineA9, while interchanging its endpoints, if and only if its axis is orthogonal to Ao.

If g is elliptic, then the point of intersection, x of A. and A, is fixed by g. Sinceit is also fixed by b, it lies on B. If A9, A and B all meet at a point x, then x is afixed point of g e H', so g is elliptic.

B.3. Proposition. Let g be a parabolic element of MI, with fixed point z, and let Abe a hyperbolic line with one endpoint at z. Then there is a hyperbolic line B # A,where B also has one endpoint at z, so that g = boa, where a is the half-turn aboutA, and b is the half-turn about B.

Proof. We can assume that g(z) = z + 1, and that A has its other endpoint at 0;i.e., a(z) = - z. Then g o a(z) z + 1, which is a half-turn about the line withendpoints at 1/2 and oo.

B.4. Proposition. Let A and B be hyperbolic lines in H3 that do not have a commonendpoint. Then there is a unique hyperbolic line C orthogonal to both A and B.

Proof. Normalize so that the half-turn a about A is the transformation a(z) _ -z.Let b be the half-turn about B; write b in the form (see I.B.4)

_ 1 - i(x + y) 2ixyb x - y -2i i(x+y)

Interpret the matrix

g = ab - ba =1 (0 -4xy)

x-y 1\-4 0 J

as an element of faro, and observe that g is a half-turn that interchanges theendpoints of both A and B. It follows that g preserves both A and B, so the axisof g is orthogonal to both.

To prove uniqueness, suppose C and C are hyperbolic lines orthogonal toboth A and B. Let g, g' be the half-turns about C, C, respectively. Then h = go g'preserves the endpoints of both A and B. Since h has four fixed points on thesphere at infinity, h = 1.

BS. Proposition. Let A and B be hyperbolic lines in H3, and let a and b be half-turnsabout A and B respectively. Set g = boa.

V.C. The Finite Groups 87

(i) If A and B do not intersect in H 3, then g is loxodromic, and the axis of g isorthogonal to both A and B.

(ii) If A and B have one common endpoint z on aH3, then g is parabolic withfixed point z.

(iii) If A and B intersect at x e O-03, then g is elliptic, and the axis of g passesthrough x and is orthogonal to both A and B.

Proof. In cases (i) and (iii), let C be the common perpendicular to A and B. Sincea and b both preserve C, while interchanging its endpoints, g preserves C andboth its endpoints; hence C is the axis of g. In case (iii), g(x) = x, so g is elliptic. Incase (i), g acts as a translation along C, and has no fixed point on it; in this case,g is loxodromic.

For case (ii), normalize so that the endpoints of A are at 0 and oo, and so thatthe endpoints of B are at 1 /2 and oc. Then a(z) = -z, b(z) = -z + 1, andg(z)=z+ 1.

V.C. The Finite Groups

C.I. In this section we classify the Kleinian groups with no limit points. We startwith the obvious remark that a Kleinian group has empty limit set if and onlyif it is finite.

C.2. Theorem. Let G be a subgroup of RA in which every nontrivial element is elliptic.Then G has a fixed point in I-Os.

Proof. Let f and g be elements of G. We already know all possibilities for whichf and g commute; i.e., either f and g lie in a common cyclic subgroup of G, or J'and g are both half-turns, where each interchanges the fixed points of the other.In either case, f and g share a common fixed point in 0-03. From here on, weassume that f and g do not commute.

Since [f g] is not parabolic, A f and A, do not have a common endpoint onthe sphere at infinity (see I.D.4). Let A be the common perpendicular to Af andA9, and let a be the half-turn about A. Write f = boa, and g = a o c, where b andc are half-turns about the lines B and C, respectively. Since f, g, and fog = h o care all elliptic, the three lines A, B and C either meet at a point, or they form atriangle. If they all meet at a point x, then x is a fixed point of <f, g>. We nowassume that they form a triangle; let P be the (hyperbolic) plane containing thistriangle.

The half-turns a, b, and c all preserve P, hence f and g both preserve P.Normalize so that G acts on B3, and so that P = 032. Consider the commutator[f,g] = boaoaocoaobocoa = (bocoa)2. Each of the half-turnsa,bandcactson P as a reflection about its axis; hence each of these half-turns reverses


orientation on 1.12. The product b o c o a is thus an orientation reversing elementof P2 whose square has finite order; i.e., b o c o a is an orientation reversing squareroot of [f, g]. By I.E. an orientation reversing square root of a elliptic elementof vN has no fixed point on C, so can preserve no disc. We conclude that A, Band C all meet at a point x.

Since each of the triples of lines (A, B, C), (A, B, A1), (A, C, AB) meet at a point,they all meet at the same point x, the common fixed point of f and g.

We have show that if f and g are elements of G with distinct axes, then theaxes of f and g intersect at a point x in H3; of course the axis of every elementof < f, g> passes through x.

We next remark that the axes of the elements of <f,g> are not all coplanar.If f and g are both half-turns, then the axis of fog is orthogonal to the axes ofboth f and g; if say g is not a half-turn, then the axes of f, g, and g o f o g-' arenot all coplanar.

Now let It be some element of G which is not in < f g>. By the above, Ahintersects the axis of every element of <f, g>. Since these axes are not all coplanar,this is possible only if Ah passes through x. O

C3. Corollary. IfG is a subgroup of P3+ in which every element has finite order,then G is conjugate in P3+ to a subgroup of 03+

C.4. Theorem. A discrete subgroup of RA is finite if and only if every element hasfinite order.

Proof. If every element of G has finite order, then by C.2, G is conjugate in L' toa subgroup of 03. By IV.E.3, a discrete subgroup of 03 is necessarily finite.

C3. We classify the finite Kleinian groups by writing down a collection of suchgroups, and then showing that every finite Kleinian group is conjugate to one ofthese. We start with the finite cyclic groups. For every integer v, there is anessentially unique cyclic group of order v: <z -> e2"t'"z). Every other cyclic groupof order v is conjugate to this one. It has basic signature (0, 2, v, v).

C.6. The dihedral groups, which can be defined as those noncyclic finite Kleiniangroups containing a cyclic normal subgroup, are as follows. For each integerv > 2, let H be the cyclic group of order v with fixed points at 0 and oo, and letG be the group generated by H and b(z) = 1/z. Since b normalizes H, the elementsof G - H are precisely the half-turns: z - e2iimI"/z, m = 0, ..., v - 1. ThusIGI = 2v. This group G, or any conjugate of it, is called the v-dihedral group.

Construct a fundamental polygon for G acting on S2 as follows. Let zt, z2,z3 be the points e- "'i", 1, e`1"; these are three successive fixed points of elementsof G - H on the unit circle 5'. Draw (Euclidean) line segments s, from 0 to zl,st from 0 to z3, and let s2 be the arc of 5' from zt through z2 to z3. Observethat under stereographic projection, these three sides project onto geodesics on

V.C. The Finite Groups 89

Fig. V.C.I

S2. Note that a(z) = e2it1'z maps s, onto s'1; also b has a fixed point at z2, andmaps s2 onto itself (see Fig. V.C.

It is easy to see that the construction above yields a fundamental domain Dfor G (one does not need Poincare's theorem in this simple case). We fold togetherthe sides of D to obtain that G has basic signature (0, 3; 2, 2, v) we also note thepresentation G = <a, b: a' = b2 = (a o b)2 = 1).

C.7. There is an important difference between the odd and even dihedral groups.If A is the axis of an element of G - H, then there is a element of H thatinterchanges the endpoints of A if and only if v is even. That is, if v is odd, eventhough there are two distinct branch points of order 2 on Q/G, there is only oneconjugacy class of elements of order 2 in G.

CA The last set of finite groups are the groups of motions of the regular solids.Let Q be a regular solid, and let G be the orientation preserving half of its groupof motions. One easily sees that G is generated by a primitive rotation a, keepinga face F invariant, and by a rotation b, which keeps an edge E of F invariant.One also easily sees that a "fundamental domain" for the action of G on thesurface of Q is a triangle D with one vertex at the center of F, and the other twovertices at the endpoints of E. Then a identifies the two sides of D emanating fromthe center of F, and b keeps E invariant and has a fixed point in the middle. Alsoa o b fixes one of the endpoints of E.

We use radial projection from the origin to project this triangle onto S2, thisgives us a fundamental polygon for G on S2.

We leave it to the reader to read off the following information.(i) If Q is a tetrahedron, then G has the presentation: G = <a, b: a3 = b2 =

(a o b)' = 1), and G has signature (0, 3; 2,3,3).(ii) If Q is a cube, G has the presentation: G = <a, b: a` = b2 = (a o b)' = 1),

and G has signature (0, 3; 2, 3, 4).


(iii) If Q is a dodecahedron, then G has the presentation: G = <a, b: a3 = b2 =(a o b)' = 1 >, and G has signature (0, 3; 2, 3, 5).

The list above does not include the octahedron or icosahedron because, asgraphs, the octahedron is dual to the cube, and the icosahedron is dual to thedodecahedron. A graph and its dual both have the same group of motions.

The group of motions of the tetrahedron has only one conjugacy class ofcyclic subgroups of order 3. This is similar to the case of the odd dihedral groups.For orientable surfaces, these are the only branched universal covering groupsto exhibit this behavior.

C.9. Theorem. Let G be a finite subgroup of M. Then G has basic signature (0, 2; v, v),v < oo, or (0, 3; 2, 2, v), v < oo, or (0, 3; 2, 3, 3), or (0, 3; 2, 3, 4) or (0, 3:2,3,5).

Proof. Normalize G so that neither 0 nor oo is a fixed point of any non-trivialelement of G. Enumerate the elements of G as {g,,,}, m = I, ..., n, and considerthe function

.f(z) _ fl g,n(z)

It is immediate that f is automorphic; that is, fo g(z) = f(z), for all g e G.Hence f defines a holomorphic map f Q(G)/G -. C. The equation f(z) = 0 hassolutions exactly at the n distinct points: g-'(0). and these are all G-equivalent;hence /takes on the value 0 exactly once. Since a holomorphic map betweencompact Riemann surfaces takes on each value equally often, f is a homemor-phism. We have shown that S2(G)/G has genus 0.

Next consider the zeros and poles of the derivative f'. There is a double poleat each finite pole of f, and f and f' are both regular at oo. Hence f' has 2n - 2poles.

Since f is injective, the zeros of f' occur exactly at the elliptic fixed pointsof G; at a point z where the projection map has branch number v (i.e., (Stab(z)I = v), f' has a zero of order v - 1. Let z1, ..., zk be a complete list ofnon-equivalent elliptic fixed points of G, and let v, be the order of Stab (zm).

Since there are n/vn distinct points in the orbit of z., the number of zeros off' is

k k

Y 1)n/vm = Y n(1 - 1/vn,).M=1 n.=1

Of course f' is rational, so the number of zeros is 2n - 2, the number of poles.Since n > 0, and each v,n z 2, k 5 3. Note that k = 0 implies that G is trivial;

k = I is impossible, for the sphere with one puncture is simply connected; andk = 2 can occur only if v, = v2, in which case G is cyclic, and n = v, = v2.

If k = 3, then I /v, + 1 /v2 + 1 /v3 = 1 + 2/n. Then, assuming that v, 5 v2 5 v3,the only possibilities are that (V,, v2, v3) = (2, 2, v) and n = 2v, or (v,, v2, v3) =(2, 3, 3), (2, 3, 4), (2, 3, 5) with n = 12, 24, 60, respectively.

V.D. The Euclidean Groups 91

C.10. Theorem. Let G be a non-trivial f finite Kleinian group. Then G is either cyclicor dihedral, or G is conjugate in RA to the group of motions of a regular solid.

Proof Comparing the lists of signatures of the groups constructed in C.5-6, 8with those that appear above, we see that given the finite Kleinian group G, thereis a group H constructed above with the same signature.

Since the action of G on ft is triply transitive, we can rind a conformalhomeomorphism P. Q(G)/G -+Q(H)/H where f preserves special points, withtheir markings, in both directions. Since G and H are both branched universalcovering groups (in particular, they define topologically equivalent coverings ofthe same marked Riemann surface; in this case a sphere with two or three specialpoints), we can lift f to a conformal homemorphism J: C - C, where finduces an isomorphism of G onto H. Of course, every conformal homeomorph-ism of C is in M. 0

C.11. The non-cyclic finite groups all have signature (0, 3; v1, v2, v3). These groupsare called the finite triangle groups. We also sometimes specify such a group asa (v1, v2, v3)-triangle group.

V.D. The Euclidean Groups

D.I. In this section we classify those Kleinian groups with exactly one limitpoint; by C.2., each such group contains a parabolic element. Let G be such agroup, normalized so that the one limit point is at oo. Since G is discrete, itcontains a minimal translation j. We also normalize so thatj(z) = z + 1: that is,if z -, z + t is any other parabolic element of G, then IT 12' 1.

Let J be the subgroup of G consisting of all the parabolic elements of G. SinceG has only the one limit point, J is commutative; since G is discrete, the rankof J is either one or two (see IV.J.15). We define the rank of G to be the rank of J.

Every non-trivial element of G is either elliptic or parabolic, and every ellipticelement of G has a fixed point at oe. Hence every element of G is a Euclideanmotion. For this reason, the discrete groups with one limit point are alsosometimes called Euclidean groups.

D.2. If G is parabolic cyclic, then G = {z - z + n, n e7L}; and G has basic signa-ture (0, 2; oo, ao).

D.3. If J has rank 1, and J 0 G, then any element a of G - J must preserve theinvariant lines of J = Q>; such an element can only be a half-turn with one fixedpoint at ao. If a and b are two such half-turns then, by B.5, a o b e J. The Euclideanline through the finite fixed points of a and b is invariant under a o b, so this linemust be parallel to the real axis. We pick one such a and further normalize so


-1/2 0 1/2

Fig. V.D.1

that the other fixed point of a is at 0. The minimality of j = a o b ensures that thefixed points of the elements of G - J occur precisely at the half integers. We easilyconclude that G = {z - ± z + n, n e 7L}, that G has the presentation <a, b: a2 =b2 = (a o b)' = 1 >, and that G has basic signature (0, 3; 2, 2, oo) (see III.G.8).

We remark that this group G or any conjugate of it, is called both the infinitedihedral group and the (2, 2, oo)-triangle group.

A fundamental polygon for G is given by the half strip {z I I Re(z)I < 1/2,Im(z) > 0}; see Figure V.D.I.

D.4. If J = G, and J has rank 2, then G is one of the usual elliptic groups generatedby z -+ z + 1, and z -+ z + z, where IM(T) # 0. In this case, G has the presenta-tion <a, b: [a, b] = 1 >, and basic signature (1, 0).

D.5. Now assume that J has rank 2, and that J # G. Let a be some elliptic elementof G; we can assume without loss of generality that a(0) = 0. Then j o a o j-' isalso elliptic with fixed point at 1, and b = a-' o j o a o j-' is parabolic with b(l) =a-'(1), a point on the unit circle. Since the minimum translation length for aparabolic element of G is one, the (Euclidean) distance, I 1 - b(1)I >- 1. It followsthat the order of a is at most six. Similarly, a cannot have order five, since11 +e 4051 < 1.

D.6. We start with the case that G contains an element of order six, and normalizeso that a(z) = e"y3z is in G. Then G also contains the elements a3(z) = -z, andb(z) = joa3(z) = -z + 1, a half-turn with fixed point at 1/2. We next constructa fundamental polygon for the group Go = <a, b>. Let D be the (Euclidean)

-"t16/.. 3 (see Figurepolygon with vertices, in order, at 0, 1/2, and e-"1'1.,,13-V.D.2), and let the sides be labeled, in the same order, as si, s2, s2, and sl. Notethat a(sl) = s;, and that b(s2) = S2'-

Using Poincare's theorem, we see that Go has the presentation <a, b: a6 =b2 = (boa)3 = 1>; folding together the sides of D, we see that Go has basicsignature (0, 3; 2, 3, 6).

V.D. The Euclidean Groups 93

Fig. V.D.2

e"6/17

e -nve/F3

Fig. V.D.3

Since Go contains j = 6 o a3 and a o j o a-' (z) = z + e"'I', Go contains a rank2 purely parabolic subgroup generated by two independent translations, both oftranslation length 1. Since no element of J can have translation length less thanone, Go contains J. The distance between finite fixed points of half-turns mustbe at least 1/2; since every point of D is at distance at most 1/2 from the point1/2, every half-turn in G is already in Go. Similarly, if we had an element c of or-der 3 in G - Go, then we could assume that the finite fixed point x of c lies in D,and we could write c(z) = e"'1'(e""'z + a). Compute a-2 o c. This is a translationby ae-'13; hence I a I z 1. Then I x I = Iai/f 1/f. contradicting the assump-tion that x lies in D, and is not one of the vertices of D. We conclude that G = Go.

D.7. We next take up the case that G has an element of order 3, but contains noelement of order 6; in particular, we assume that in addition to j(z) = z + 1, Gcontains the element c(z) = e2"t/3z. As in the preceding case, we set Go = <cj),and construct a fundamental polygon for Go.

Let D be the polygon with vertices, in order, at 0, e"/6/ f, 1, and e--1/6/,/3-;label the sides of D, in the same order, as s,, si, s2, and s2 (see Figure V.D.3).


Note that a = j o c is elliptic of order 3, has a fixed point at ?"6/f3, and mapss, onto s,. Also b = j o c-' is elliptic of order 3, has a fixed point at e-"f/f,

and maps s2 onto s2. We conclude from Poincare's theorem that Go has thepresentation <a, b: a3 = b3 = (a o b)' = 1 >; folding together the sides of D, we seethat Go has basic signature (0, 3; 3, 3, 3).

As in the preceding case, observe that G - Go contains no parabolic elements.We saw in D.6 that the distance between fixed points of distinct elliptic elementsof order 3 must be at least 1/-,/3-, so there are no elements of order 3 in G - Go.If there were an element of order 2 in G - Go, write it as d(z) = i(iz + a), andobserve that a o d has order 6.

D.8. We next take up the case that G contains an element of order 4; normalizeso that a(z) = iz lies in G. Set b = j o a2; b is a half turn with fixed point at 1/2.As in the preceding cases, we construct a fundamental polygon for Go = <a, b>,and then show that Go = G. Our polygon D has vertices at 0, (1 + i)/2,1/2, and(1 - i)/2. The sides of D are labelled in order as s;, s2, s2, s, (see Figure V.D.4).Note that a(s,) = s,, and b(s2) = s2. Using Poincare's theorem, we see that Gohas the presentation <a, b: a" = b2 = (a o b)° = 1 >; fold together the sides of D, tosee that Go has basic signature (0, 3; 2,4,4).

We already know both Euclidean groups that contain an element of order 3,and neither of them contains an element of order 4; therefore, G - Go contains noelement of order 3. Also Go contains both j and a o j o a-', so G - Go can containno parabolic element. As above, G - Go can contain no half-turns, because thedistance between finite fixed points of half-turns in G must be at least 1/2.

D.9. Finally, we take up the case that every elliptic element of G has order 2.Observe that if we start with any group J of signature (1,0); i.e.,

Fig. V.D.4

V.E. Applications to Non-Elementary Groups 95

0

T/2 Sj 11.TI/2 s3 1.T/2

S, S"1/2

Fig.'V.D.5

1

J = {z-+z+n+mrlIm(r)>0,n,me7L},and we adjoin b(z) = -z to J, we obtain the group Go = {z -+ ±z + n + mr}.

If we start with the standard fundamental parallelogram for J with verticesat 0, 1, 1 + r, and r, then the parallelogram with vertices at 0, 1, 1 + r/2 and r/2is a fundamental polygon for Go. The vertices of this fundamental polygon areat 0, 1/2, 1, 1 + r/2, (1 + r)/2, and r/2. We label the sides of this polygon in orderas s,, si, Si, s3, s3, and s2 (see Figure V.D.5). Let c = job; c is a half-turn, withfinite fixed point 1/2, mappings, onto s,. The sides s2 and s2 are paired by j, anda(z) = - z + I + r maps s3 onto We can also write a = j o k o b, where k(z) =z + r is the other standard generator of J. It is easy to see that Go has thepresentation <a, c, j: a2 = (j o a)2 = c2 = (j-t O C)2 = I>, and that Go has basicsignature (0, 4; 2,2,2,2).

The presentation above is not the standard one. Make the substitutionj = cob to obtain the usual presentation:

Go = <a,b,c: a2 = h2 = c2 = (cohoa)2 = 1>.

It is clear that we cannot adjoin a new half-turn to Go without also adjoininga new translation to J. Hence Go = G.

D.10. We have exhausted the list of Euclidean groups. Every Kleinian group withexactly one limit point is conjugate in M to one of the groups listed above.

The groups with basic signature (0; 3; v1, v2i v3), 1/v1 + 1/v2 + I/v3 = 1, arecalled the Euclidean triangle groups (the infinite dihedral group is usually in-cluded among the Euclidean triangle groups). As with all triangle groups, we alsosometimes refer to such a group as the (vt, v2, v3)-triangle group.

V.E. Applications to Non-Elementary Groups

E.1. Proposition. Let G be a Kleinian group with at least two limit points, then Gcontains a loxodromic element.


Proof. By C.4, G cannot contain only elliptic elements. If G has no loxodromicelements and if all the parabolic elements of G have a common fixed point, thenG is conjugate in Iy0 to a group of Euclidean motions (see V.D). Hence, if Gcontains no loxodromic elements, then G contains a parabolic element f withfixed point at x, and a parabolic element g with fixed point at y # x. NormalizeG so that x = oo and y = 0. An easy computation using the normal form (1.1)shows that either fog or fog-' must be loxodromic.

E.2. Proposition. Let G be a non-elementary Kleinian group, and let x e A. Then thetranslates of x are dense in A.

Proof. G contains a loxodromic element f. If f(x) = x, then there is a limit pointy which is not fixed by f. Then by II.D.2, there is a fixed point z of f, and thereis a sequence of elements {g,,, } of G, with g.(z) - y. For every g e G, the fixedpoint set of f and g o f o g' either agree or are disjoint (II.C.6). Hence there is anelement g e G, so that the fixed points of g ofo g-' are distinct from those of f; inparticular, there is a translate of x distinct from x. Of course we have the sameresult if f(x) x. The desired result now follows from II.D.2.

E3. We obtain the following as an immediate corollary.

Proposition. The limit set of a non-elementary Kleinian group is the closure of theset of loxodromic fixed points.

E.4. Proposition. Let G be a non-elementary Kleinian group, and let E be a non-empty G-invariant closed set. Then E contains A.

Proof. It follows from II.D.2 that the translates of any point of 0 are dense in A;by E.2, the translates of any point of A are dense in A. Now let x be any point ofE. Then the set of all translates of x is contained in E, so the closure of this set,which contains A, is also in E.

E.S. The following is a useful generalization of the preceding.

Proposition. The set of loxodromic fixed points of a non-elementary Kleinian groupis dense in A x A (i.e., for every pair of points x, y in A, there is a sequence ofloxodromic elements of G, where the attracting fixed points approach x, and therepelling fixed points approach y).

Proof. Normalize G so that oc a Q. There is a sequence of loxodromic elements ofG whose attracting fixed points approach x, hence there is a sequence {gm} ofloxodromic elements of G, so that both fixed points of g11, approach x. If x = y,we are finished; if not, then as above, there is a sequence of loxodromic elements

where both fixed points of h,, approach y.

V.E. Applications to Non-Elementary Groups 97

For in sufficiently large, the isometric circles of gm and gm' are disjoint fromthose of hm and hm'. Observe that fm = h;' o gm maps the outside of the isometriccircle of gm into the inside of the isometric circle of hm. Hence the attracting fixedpoint of fm lies inside the isometric circle of hm, and the repelling fixed point offm lies inside the isometric circle of gm. As in increases, the radii of these isometriccircles tends to zero; hence the fixed points of fm approach x and y.

E.6. We need a concept of convergence of Kleinian groups for the next applica-tion. Let G be a finitely generated Kleinian group, and let {cpm} be a sequence ofhomomorphisms of G into M. We say that ipm converges algebraically to thehomomorphism gyp: G - M, if there is a set of generators (g1,. .. , gk } for G, so that(pm(gj) - 4p(gj),.1 = 1, ... , k.

Theorem (Chuckrow [20]). Let G be a non-elementary Kleinian group, and let{lpm} be a sequence of type-preserving isomorphisms of G into M. Suppose that foreach in, rpm(G) is Kleinian, and that c& converges algebraically to ip: G - . Thencp is an isomorphism.

Proof. It suffices to show that for every non-trivial g e G, cp(g) -A 1. Assume firstthat g is loxodromic. Since G is non-elementary, there is an element h e G, so thatg and f = h o g oh' have distinct fixed points. By Jorgensen's theorem (II.C.7),µm = I tr2((pm(g)) - 41 + I tr([(pm(f ), (pm(g)] - 21 >: 1, for every in. The entries inthe matrix for g are polynomials in the entries in the matrices of the genera-tors of G, hence, cpm(g) - qp(g). If (p(g) = 1, then

µ = Itr'((P(g)) - 41 + Itr[w(f),q(g)] - 21 = 0.

Except for the absolute values, this function is also a polynomial in the entriesof the generators, hence um -, p.

If g is elliptic, then tr2((pm(g)) is constant, and not equal to four, so there isnothing to prove in this case.

If g is parabolic, then, as above, there is some h e G, so that g and f =h o g o h-1 have distinct fixed points. If 9m(g) -* 1, then 4pm([g, f *]) I for allintegers k. Normalize g so that g(z) = z + 1, and so that f has its fixed point at0. Write f(z) = z/(rz + 1), and compute tr2 ([g, f'`]) = (2 + k2t2)2. Hence [g, fk]is loxodromic for k sufficiently large, and we can apply the argument above.

0E.7. Proposition. Let G be a non-elementary Kleinian group which has n z 2 invar-iant components. ?hen n = 2.

Proof. Let g be a loxodromic element of G, and let dm be an invariant component.Let wm be a simple <g>-invariant path in J. (such a path is easily constructedby choosing a point zm in A. and choosing a simple path vm from zm to g(zm);then wm is the union of the <g>-translates of vm). The union of w1, w2 and thefixed points of g is a simple closed curve C that divides 4' into two regions. Each


of these regions contains limit points that lie on the common boundary of A,and 42. By E.5, there is a loxodromic element h e G whose fixed points areseparated by C. We conclude that any <h>-invariant curve must cross C. Thefixed points of h are distinct from those of g, so the point of intersection occursin either A, or 42. Since any invariant component A contains an <h)-invariantcurve, A = A or A = 42.

E.8. Proposition. Let G be a Kleinian group, and let .4 be a component of G. EitherA is G-invariant, or there is another component 4' so that A U X is G-invariant, orthere are infinitely many distinct translates of A in Q.

Proof If G is elementary, then G has only one component; hence we may assmethat G is non-elementary. Suppose there are only finitely many translates of A,call them A,, ..., A,,. Then for each m, Stab(A,,) is of finite index in G. LetH= n Then H, being of finite index in G, is non-elementary. Since Hkeeps each of the components A,,._ 4, invariant, k = 2.

E.9. If A is a connected component of Q(G), then Stab(A) is called a componentsubgroup of G.

Proposition. Let G be a Kleinian group with finitely many components. Then G hasat most two components.

Proof. If G is elementary, then G has only one component; hence we may assumethat G is non-elementary. Since G has only finitely many components, the stabi-lizer of any one component is of finite index in G. As above, let H be the inter-section of all the component subgroups of G. Then H is of finite index in G, andH keeps every component of G invariant. Since A(H) = A(G), the componentsof G are just the components of H, and H has at most two components.

E.10. Proposition. Let G be a non-elementary Kleinian group and let C be thenormalizer of G in M. Then C is Kleinian, and Q(C) = SQ(G).

Proof. We first show that C is discrete; assume not. Then there is a sequence {gm}of elements of C tending to the identity. If g is any loxodromic element of G, thenh. = g,,, o g o g,-' is also in G, and h. -+ g. Since G is discrete, we must have thatfor m sufficiently large, h. = g; then g have the same fixed points. SinceG is non-elementary, it contains many loxodromic elements with distinct fixedpoints. Hence for m sufficiently large, g,,, = 1.

Suppose there is a point x e Q(G), where x is a limit point of C, Then, regardingC as acting on 63, there is a sequence {gm} of elements of C, with g,,(0) - z. ByIV.G.9, there is a point y, and a subsequence {gm}, so that g.(z) - x uniformlyon compact subsets of the complement of y. In particular, there is a point w e A(G),with g.(w) - x. Since every element of 0 stabilizes A(G), this is impossible.

V.F. Groups with Two Limit Points 99

E.11. The next application concerns the action of a Kleinian group on hyperbolic3-space.

Proposition. Let G be a discrete subgroup of M. Then H'/G is a 3-manifold.

Proof. Let x be some point of H3. Since G is discrete, Stab(x) is finite, and thereis a precisely invariant hyperbolic ball B about x. Normalize so that G acts onB', and so that x = 0. Then we can write B = { y I I y1 < r}.

The elements of H = Stab(0) commute with the dilations x -' A.x, .t > 0; hencethe actions of H on the spheres S, = {yI IYI = r'} are all equivalent. We canregard B as S, x [0, 1], with S, x {0} identified to a point. Then B/H can beregarded as (S,/H) x [0, 1], with (S,/H) x {0} identified to a point. We look inV.C. at the list of signatures for finite groups, and observe that they all have genuszero. Hence S,/H is a 2-sphere; so B/H is a 3-disc. We conclude that H'/G is a3-manifold. Q

V.F. Groups with Two Limit Points

F.I. In this section we classify the Kleinian groups with exactly two limit points.We observed in E.1 that such a group G necessarily contains a loxodromicelement. Normalize G so that j(z) = t2z, I tI > 1, generates a maximal loxodromiccyclic subgroup J of G. It is easy to see that if h(z) = s2z is any other loxodromicelement of G, then Isl = Itl', for some integer m. We set H = Stab(0) = Stab(oo).Since G has only the two limit points, G = Stab({0, oo 1). Clearly J c H c G.

F.2. We already know all possible basic signatures for a group G with two limitpoints. The universal covering space of Q(G) = C - {0} is the complex plane.Hence there is a universal covering group C, acting on C, so that C/C isconformally equivalent to Q(G)/G. Of course, a conformal map of C lies in M;hence C is a Kleinian group with one limit point. We listed all the signatures forsuch groups in V.D; they are (0, 2; oo, oo), (0, 3; 2, 2, oc), (0, 3; 2,3,6), (0,3; 3,3,3),(0, 3; 2,4,4), (0,4; 2,2,2,2), and (1, 0). In fact, as we will see below, only the lasttwo actually occur for groups with two limit points.

F.3. If J = G, then G is loxodromic cyclic, and the obvious fundamental domainfor G is the annulus D = { z I I t I -1 < I z I < I t I }, where j identifies the two sides ofD. Folding together the two sides of D, we see that G has basic signature (1, 0).

F.4. We next take up the case that J # H = G. Choose an element h in H - J;h(z) = s2 z. Then l s l = I t l'°, for some integer m. Replace h by j-'o h, so that Is l = 1;i.e., we can assume that h is elliptic, with fixed points at 0 and oc. We can alsoassume without loss of generality that <h> is a maximal elliptic cyclic subgroup


1

Fig. V.F.1

1

Fig. V.F.2

Itl2

of H, and that h is a geometric generator of <h>; i.e., h(z) = e2"tlgz. Every elementof H commutes with both j and h; since these generate maximal cyclic subgroupsof H, G = H = Q, h>. H, or any conjugate of it, is called 7L + 7Lg, which accuratelydescribes its group theoretic character; i.e., G = Q, h: [j, h] = hg = I>.

We choose a fundamental domain D for G to be the "rectangle": { 1 < IzI <t2 1, 0 < arg(z) < 2n/q}. If j is hyperbolic, then the opposite sides of this rectangle

are paired by j and h, and one easily sees that G has basic signature (1,0) (seeFigure V.F. 1). Note that the relation [j, h] = 1 can be read off from the identifica-tions at the vertices of D, but the relation h9 = 1 cannot.

If j is loxodromic but not hyperbolic, then the pairing of the sides is somewhatmore complicated. The sides {arg(z) = 0}, and {arg(z) = 2n/q} are still paired byh. We write t2 = where 0:5 0 < l/q. Then h-"oj(l) = y is a pointon the boundary of D. Let x =' o h"(It2Ie2i'1g). Each of the points x and y splitsthe circular side on which it lies into two arcs; one pair of these four arcs isidentified, as we have already observed, by h-" o j; the other pair is identified byh" o f. In this case D has six sides, with each pair of opposite sides identifiedby a element of G (see Figure V.F.2). This again is a torus; i.e., G has basicsignature (1, 0).

V.F. Groups with Two Limit Points 101

As above, the identifications at the vertices of D yield the information that jand It commute, but do not yield the relation h9 = 1.

FS. We next take up the case that J = H # G, and we look at an element g inG - J. Since G has only the two limit points and g does not fix them, it mustinterchange them. It follows that every element of G - J is a half-turn whichinterchanges 0 and oo. The product of any two such elements fixes 0 and oo,hence J is of index 2 in G.

Normalize G further so that g has a fixed point at 1; then g(z) = 1/z. Since Jhas index 2 in G, G = <J, g>. Let k(z) = j o g(z) = t2/z. Note that k has fixed pointsat ± t.

Since G contains J as a subgroup of index 2, we expect a fundamental domainfor G to be, in some sense, half of a fundamental domain for J. Choose D to bethe annulus (I < I z i < I t I } (observe that the closure of D Uj(D) is the closure ofthe fundamental domain for J described in F.3).

Note that ± 1 are vertices that divide Iz1 = 1 into two sides; these are identi-fied by g. Also ±t are vertices that divide { I zI = I t I } into two sides that areidentified by k. It follows that G has basic signature (0,4;2,2,2,2) (see FigureV.F.3).

Finding a presentation for G is also easy. Extend the sides of D into U-l'; thatis, draw the hyperbolic planes whose boundaries are the sides of D. It is clearthat these bound a polyhedron where the sides are identified by g and k. Weobtain the presentation G = <g, k: g2 = k2 = 1>.

For purposes of identification, this group is known as Z2 * Z2, which alsoaccurately describes its group theoretic character (G * H is the free product of Gand H).

F.6. We now consider the general case, where J 96 H # G. Since G # H, Gcontains the group, which we now call Go, constructed in F.5. Also, by assump-tion, there is some element h in H - J; clearly h e G - Go. As in F.4, we can

Fig. V.F.3


Fig. V.F.4

assume that h has the form h(z) = e2w'Igz, and that <h> is a maximal cyclicsubgroup of G. Let G' = <Go, h>; we will show below that G' = G.

Consider the region D = (I < Izl < Itl, -n/q < arg(z) < n/q). The sides{argz = ±n/q are identified by h. The point 1 is a fixed point of g(z) = 1/z. Itdivides the side { Izl = 1) on 8D into two arcs that are identified by g. Theidentifications on { I z I = I t 1) depend on t 2 = it 2 le2i '(nlq+e). If j is hyperbolic (i.e.,

t > 0), then k = jog has a fixed point at t, and interchanges the two halves of{IzI = t}. Similarly if 0 = 0, then h-"ojog fixes the point It!, and interchangesthe two halves of { Iz I _ It!) on D. In the general case, where B # 0, the arc{ I z I = I t I ) on OID has two elliptic fixed points on it; these are fixed points of theelements k = h-"ojog, and k' = h-"-'ojog. These two fixed points divide thearc { I z I = I t I } into four sides that are pairwise identified by the two half-turns.The four sides are defined by the vertices v, on {arg(z) = -n/q}; v2 is the fixedpoint of k'; v3 = k'(v,) = k-'(v5); v4 is the fixed point of k; and v5 lies on{arg(z) = n/q}. Folding together all the sides of D, we see that G' has basicsignature (0,4;2,2,2,2) (see Figure V.17.4).

As in the preceding case, we can extend the sides of D into H3 to obtain afundamental polyhedron for G' from which we can read off a presentation. Exceptfor the relation hq = 1, the relations can all be read off from the vertices of D;this other relation appears on the edge z = 0 in H3, which has no counterpartin D. We obtain G' = <g, h, k: g2 = (g o h)2 = hq = k2 = (k o h)2 = 1>.

We might still have that G is a non-trivial extension of G'. Assume that thereis a transformation f in G - G'. We have already assumed that G contains j asa minimal loxodromic element, and that it contains h as a minimal rotation. Theonly possibility left is that f is a half-turn which interchanges 0 and 00, so g ofhas fixed points at 0 and oe; hence g of lies in the subgroup of G' generated byj and h. We conclude that fE G'.

The group G constructed in this section, or any group conjugate to it, is calleda double dihedral group. The reason for this is the following. Let H, be thesubgroup of G generated by g and h, and let H2 be the subgroup generated by hand k. Then H, and H2 are both q-dihedral groups. These two subgroups

V.G. Fuchsian Groups 103

generate G, in fact, G is the free product of H, and H2 with amalgamatedsubgroup <h> (see VIII.C.1).

V.G. Fuchsian Groups

G.1. One needs an understanding of the theory of Fuchsian groups in order tounderstand Kleinian groups; we will present here only as much of that theory asis needed for the subsequent development. Our primary concern is with defini-tions and notation; we present proofs of only those facts which, in the authors(admittedly limited) experience, are difficult to find in the literature (see alsoVI.A.6-7).

A Kleinian group G is Fuchsian if it keeps invariant some circular disc U.The boundary of U is called the circle at infinity; it is clear that the circle atinfinity contains the limit set of G.

We can regard U as being H2, so that a Fuchsian group is a discrete subgroupof L2+ = PSL(2, R), or we can regard U as being B2, in which case our Fuchsiangroup is a discrete subgroup of P2+.

G.2. The basic signatures (0, 1; v) and (0, 2; v,, v2), v, s v2, cannot occur in aKleinian group for topological reasons. All others can and do occur. We havelisted those basic signatures which occur among the elementary groups, theothers are all basic signatures of Fuchsian groups. That is, for every basicsignature (p, it; v,,..., where

2p-2+F1/v, > 0,(by definition, l/00 = 0) there is a Fuchsian group with this basic signature (thisis an immediate consequence of the uniformization theorem and III.F.7).

G.3. Fuchsian groups are divided into two kinds. A Fuchsian group is of the firstkind if every point on the circle at infinity is a limit point; it is of the second kindotherwise.

G.4. For a Fuchsian group F of the second kind, U is not a component of F. Thecomponent of F containing U is not simply connected, unless of course, F iselementary. Hence a non-elementary Fuchsian group of the second kind is nota basic group.

We do however need one fact about Fuchsian groups of the second kind (wewill return to these groups in VIILD and X.1).

Proposition. Let F be a Fuchsian group of the second kind. Then A(F) is nowheredense in the circle at infinity.


Proof. The result is obvious if F is elementary; we assume F is non-elementary.Since 0 is open, the set of non-limit points on the circle at infinity is a set of opendisjoint arcs. The endpoint of each arc is a limit point of F. The translates of thislimit point are dense in A; hence we can approximate every point of A by regularpoints on the circle at infinity.

GS. An analytically finite Fuchsian group of the first kind is a basic group, andhas a well defined basic signature.

Proposition. Let a be the basic signature of an analytically finite Fuchsian groupof the first kind. Then a is not the basic signature of an elementary group.

Proof. If H is an elementary basic group, then °S2(H)/H is either a twice puncturedsphere, a thrice punctured sphere, a four times punctured sphere, or a torus.Conformally, there is only one twice or thrice punctured sphere; i.e., if S and 9are two such punctured spheres, then there is a conformal homeomorphismmapping S onto 9, where we can require that a given puncture on S be mappedto a given puncture on 9. Hence the branched universal covering surfaces of Sand 99 are conformally equivalent. Since the disc is conformally distinct from boththe plane and the sphere, the result follows in these cases. The groups thatuniformize the four times punctured sphere and the torus are all Euclidean, andthey all contain a rank two free Abelian subgroup. A rank two free AbelianKleinian group is necessarily parabolic (see I.D.3); such a group can preserve nodisc.

G.6. Proposition. Let F be a non-elementary Fuchsian group, and let G be anelementary group. Then F and G are not isomorphic.

Proof. Assume there is an isomorphism q : F - G. Since every elementary groupis a finite extension of an Abelian group, so is F. It is easy to see that a rank 2hyperbolic group is not discrete (the two generators must have the same fixedpoints), and a rank 2 parabolic group can preserve no disc. Hence F is a finiteextension of a cyclic group; so F is elementary.

G.7. If the Fuchsian group F has basic signature a = (p, n; v...... v.), then F hasthe presentation

F= <a,,bt,...,ap,bp,el,...,en:Il[am,bminej=ei' _ 1J.

The symbol e°° in the expression above means that e is parabolic, also, everyparabolic element of F is conjugate to some power of some e; with exponent oo.

An important fact that is almost obvious from the presentation above is thatthere is a type-preserving isomorphism between two Fuchsian groups of the firstkind if and only if they have the same basic signature.

G.B. Let F be a Fuchsian group and let S = U/F. Let °U = U fl °Q(F). Then °Uis a regular covering of its image °S. Once we have chosen base points, there is


a homomorphism 0: n, (°S) -+ F. We call an element a in n, (°S), or a loop wrepresenting a, elliptic, hyperbolic, or parabolic according as 0(z) is elliptic,hyperbolic or parabolic, respectively. It is easy to see that this classification isindependent of base point.

If a e n, (°S) is hyperbolic, then the axis A of g = 0(n.) is the geodesic in HIconnecting the fixed points of g (we will also sometimes refer to this axis as beingthe true axis). Let v be the projection of A to S; it is the geodesic on S correspond-ing to a. If F is torsion free, then v is the unique smooth geodesic, in the hyperbolicmetric, freely homotopic to w, where w is any loop representing a. If F has torsion,then the axis might pass through some elliptic fixed points.

G.9. In general, if w is a simple loop on °S, then the corresponding geodesic v isalso simple. To see this, let W be a connected component of p-' (w); W can beobtained by lifting w to a path Wo, and then interating; i.e., lift w again startingat the end point of WO, then lift w again, and so on; at the same time, lift w',starting at the beginning point of WO, then lift w-' again, and so on. The path Wis invariant under <g = O(a)>; hence it has well defined endpoints: the fixedpoints of g. Since w is simple, no translate of W under F crosses W. This meansthat the fixed points of g do not separate the fixed points of any conjugate of g;it follows that no translate of A, the axis of g, crosses A. If Stab(A) = <g> is cyclic,then we have shown that the corresponding geodesic is simple.

If Stab(A) is not cyclic, then there is a half-turn h in Stab(A) which inter-changes the fixed points of g. Then g o h is also a half-turn in Stab(A). SinceStab(A) is discrete, it is the elementary group Z2 * Z2. In this case v, the projectionof A, is a path from one ramification point of order 2 to another, and then backagain. Note that W U h(W) is the boundary of a regular neighborhood of A; hencethe original loop w is a simple loop that surrounds these two ramification points.We have shown the following.

Proposition. Let w be a hyperbolic simple loop on °S, where w does not bound a disccontaining exactly two ramification points, both of order 2. Then the geodesiccorresponding to w is also simple. If w does bound a disc containing exactly tworamification points of order 2, then the corresponding geodesic is a path that runsfrom one of these points to the other, and then back again.

G.10. The same proof also yields the following.

Proposition. Let g be a primitive hyperbolic element of the Fuchsian group F. Thereis a path Wconnecting some point z e Flt to g(z), where W projects to a simple loop,if and only if the axis A of g is precisely invariant under its stabilizer in F.

A hyperbolic element of a Fuchsian group which satisfies the above is calledsimple.


G.1 1. It is often cumbersome to work with axes with elliptic fixed points on them,and we now introduce a distinction between "axis" and "true axis". If thehyperbolic element f c- F is not simple, or if the axis off does not have fixed pointsof half-turns on it, then the axis and the true axis are the same. If f is simple andits axis has fixed points of half-turns on it, then the true axis is the geodesicconnecting the fixed points off, and the axis off is a neighboring curve A, chosenso as to satisfy the following requirements: A is <f >-invariant; A projects to asimple loop on °S; distinct axes are disjoint; and the set of axes is F-invariant. Itis clear that this can be done in many different ways; for any given Fuchsiangroup, we make such a choice.

With this definition, if the axis off is not the true axis, then the true axis hasfixed points of half-turns on it. Applying one of these half-turns to the axis yieldsa second axis for f; this second axis lies on the other side of the true axis.

G.12. Let F be a Fuchsian group acting on 82. Then Q(F) fl S' is an open subsetof §'; hence a union of arcs. Each of these arcs is called a boundary arc or arcof discontinuity of F. Each arc of discontinuity C has a boundary geodesicconnecting the two endpoints of C. The boundary geodesic A divides 82 intotwo half spaces; the half space whose boundary is C is called a boundary half space.

Proposition. Let F be a non-elementary Fuchsian group. Let C be an arc of discon-tinuity of F with boundary geodesic A and boundary half space H. Then Stab(C) =Stab(A) = Stab(H) is either trivial or hyperbolic cyclic, and H U A U C is preciselyinvariant under Stab(C).

Proof. Since the endpoints of A are limit points of F, no translate of A can havean endpoint in C. It follows that A is precisely invariant under its stabilizer. IfStab(A) is hyperbolic cyclic or trivial, then every element of Stab(A) preservesboth C and H; hence, in this case, Stab(A) = Stab(C) = Stab(H), and A U H U Cis precisely invariant under Stab(A).

Stab(A), qua Kleinian group, has either 0 or 2 limit points; hence it is eithertrivial, hyperbolic cyclic, 12, or 12 * Z2. In the former two cases, A is pre-cisely invariant under Stab(A). In the latter two cases, there is a half-turnh e Stab(A). Then the entire circle at infinity is composed of C, h(C) and the endpoints of A. It follows that F = Stab(A), contradicting our assumption that F isnon-elementary. 0

A non-trivial element of Stab(H) is called a boundary element of F.

G.13. Given a non-elementary Fuchsian group F, the convex region K(F) isdefined to be the smallest closed (hyperbolic) convex set whose (Euclidean)closure contains the limit set of F. It is easy to see that if F is of the first kind,then K(F) is the entire disc, while if F is of the second kind, K(F) is thecomplement of the union of the boundary half spaces.


G.14. An important fact about Fuchsian groups is the following.

Theorem. Let D be a Dirichlet region for the non-elementary Fuchsian group F.Then the following statements are equivalent.

(i) D has finitely many sides.(ii) F is an analytically finite Kleinian group.(iii) F is finitely generated.(iv) The hyperbolic area of K(F)/F is finite.

Also, if these conditions hold, then for every boundary geodesic A, Stab(A)is hyperbolic cyclic.

G.15. Let F be a non-elementary finitely generated Fuchsian group, and letS = H 2/F. A system of loops on S is a set { w3,. .. , wk } of simple disjoint loops on°S = (°0 (F) f1 H2 )/F satisfying the following condition. The complement in °S ofthese loops is a set of subsurfaces, { Y, , ... , Y } called the building blocks of thesystem; we require that for each Ym, tt,(Ym) is not Abelian.

A system of loops is maximal if it is not properly contained in another system;the building blocks in a maximal system are called units. Since the system ismaximal, each unit is planar (that is, has genus zero). Also each unit has exactlythree boundary components; the different boundary components are the twosides of each wm, the projection of the boundary arcs, and the special points,including punctures, on S. If F is purely hyperbolic, then each unit is a "pair ofpants"; that is, a sphere with three holes.

Proposition. Let F be a non-elementary Fuchsian group of signature (p, n;and let (w1,. ..,Wk} beamaximal system of loops onS = 0.02/F.Then k = 3p - 3 +n, and the system divides S into - X(F) = 2p - 2 + n units (X here is the Eulercharacteristic of °S).

Proof. Let S be some surface of genus p so that there is an embedding of S into9. Then we can regard the loops w,, ..., wk as being simple disjoint loops on S.There is a maximal homologically independent subset of the system, let j be thenumber of loops in this subset. Obviously j 5 p, and, since each unit has genuszero, j = p. We cut S along these p loops, so as to obtain a planar surface; theneach additional wm cuts this surface into two parts. Hence the number of units isl+k-p.

The total number of boundary components is 2k + n; since each unit hasthree boundary components, 2k + n = 3(1 + k - p). Hence k = 3p - 3 + n, andthe number of units is 2p - 2 + n. 0G.16. We also need a more complicated sets of loops. Let S be a closed markedRiemann surface. A set of loops fills up S if every loop in the set is simple; everyconnected component of its complement is simply connected; and every con-nected component of its complement contains at most one special point.


Let F be a finitely generated Fuchsian group of the first kind, where S. thecompletion of HZ/F, has positive genus. Using the fact that every non-dividingloop is hyperbolic, it is easy to construct a set of hyperbolic simple loops thatfills up S.

G.17. We say that the set of translates {gm(A)} of an axis A nests about the pointx if the axes g.(A) are all disjoint; each gm(A) separates x from g.+, (A), and (inthe spherical metric) the axes g.(A) -, x. Note that the last condition is automaticif Stab(A) and the g,,, all lie in a discrete group, for the translates of an axis mustconverge to a point.

Theorem. Let F be a finite) y generated Fuchsian group of the first kind. Let x be alimit point of F, where x is not a parabolic fixed point. Then there is a hyperbolicelement g, with axis A, and there is a sequence { gm } of elements of F, so that { g.(A) }nests about x.

Proof. If H2/F has genus 0, then there is a subgroup F of finite index, so that112/F' has positive genus. Let w,, ..., wk be a set of simple loops filling up S, thecompletion of H2/F', and let £ be the set of axes of elements lying over the wm.Since each connected component of the complement of Uwm is simply connectedon S, so also is each connected component of the complement of E. Of coursethere might be elliptic fixed points in some of these connected components, butthere can be at most one such point in each component. Similarly, there mightbe a parabolic fixed point on the boundary of some component, but again, thereis at most one such point on the boundary of any one component.

Draw the (hyperbolic) line L from some point z in H' to x. If necessary, movez a little so that L is not colinear with any of the axes in E. Since x is not aparabolic fixed point, L passes through infinitely many of the sets in the comple-ment of £, so L crosses infinitely many of the axes in £; therefore there is an axisA in £, so that L crosses infinitely many translates of A. Since each w,,, is simple,the different translates of A do not intersect; since F is discrete, the translatesof A accumulate to a point, necessarily x. 0

G.18. We conclude this section with a characterization of those Kleinian groupsthat are Fuchsian. We saw in I.B.11 that an element g of RA can keep some discinvariant if and only if tr2(g) >_ 0. Hence, for every g in a Fuchsian group,tr2(g) >_ 0.

Theorem. Let G be a non-elementary Kleinian group in which tr2(g) >_ O for allg e G. Then G is Fuchsian.

Proof. Since G is non-elementary, it contains many loxodromic, in fact hyper-bolic, elements. We normalize G so that

V.H. Isomorphisms 109

Y _Ct O10 t-1 ' t > 1, and h =

a blc d

are hyperbolic elements of G with distinct fixed points. We also normalize so thatthe product of the fixed points of h, -b/c = 1.

Since t is real, and h and go h both have real traces, a and d are both real.Also since ad - be = 1, and b = - c, we obtain that b2 = c2 is real. The fixedpoints of h are given by

a-d±((a+d)2-4)1/2

2c

Since h is hyperbolic, the numerator is real. Since c is either real or pureimaginary, both fixed points of h are also either real or pure imaginary. If theyare real, then g and h both preserve the upper and lower half-planes, while if theyare pure imaginary, then g and It both preserve the right and left half-planes.

We renormalize, keeping g fixed, so that <g, h> preserves the upper half plane;i.e., we renormalize so that <g, h> is a subgroup of PSL(2, R).

Now let

f=(ac ' d'/

be any other element of G. As above, we compare the traces of g, f, and g of toshow that a' and d' are both real. We also compute the trace of h of to obtainthat bc' + b'c is real. Then we compute the trace of g o h o f to obtain thattc'b + t-lb'c is real. Combining these last two statements, we conclude that b'and c' are both real. Hence f also lies in PSL(2, IB).

V.H. Isomorphisms

H.I. The Nielsen isomorphism theorem is an important tool in the theory ofFuchsian and Kleinian groups. We state it here in complete generality, and provea weaker version in IX.E.

Theorem. (Nielsen [78]). Let rp: G - 6 be an isomorphism between finitely gen-erated Fuchsian groups, acting on the disc U, satisfying the following conditions.

(i) g e G is parabolic if and only if (p (g) is;(ii) g e G is a minimal rotation if and only if cp(g) is; and(iii) g e G is a boundary element if and only if rp(g) is.

Then there is a homeomorphism f C - C, where fogo f'1 = tp(g), and f(U) = U.

The usual formulation of this theorem has condition (ii) replaced with thecondition that G, and hence also C, is non-elementary.

In this section we prove the analogous result for the basic elementary groups.

I to V. The Geometric Basic Groups

H.2. In general, an isomorphism cp: G -. ? is called type-preserving if conditions(i) and (ii) above are satisfied.

H.3. Proposition. Let cp: G - l; be a type-preserving isomorphism between finiteKleinian groups. Then cp is a topological deformation; in fact, G and ri are conjugatein O.

Proof. It is easy to see that two finite Kleinian groups are isomorphic if and onlyif they have the same signature. We saw in C.10 that up to conjugation in M,there is only one finite Kleinian group of each possible signature. Hence we canassume that t = G.

If G is cyclic or dihedral, choose a geometric generator j of the maximal cyclicnormal subgroup J (if G is cyclic, J = G). Then either cp(j) = j, or V (j) = j-1. Inthe latter case, let a be a half-turn interchanging the fixed points of j, and leta*: G - G be conjugation by a; that is, a,(g) = a o g o a-'. Replace cp by a* o gyp;i.e., we can assume that cp(j) = j.

If G is cyclic, we are finished. If G is dihedral, normalize G so that the fixedpoints of J lie at the poles of S2, and so that one of the fixed points of somehalf-turn in G lies on the equator; then all the fixed points of the elements ofG - J lie on the equator. Choose some element g e G - J. Let r be a rotation ofS2, where the axis of r is the axis of J, and r maps the fixed points of cp(g) ontothose of g. Replace cp by r* o cp; then cp(g) = g. Since G = Q, g), cp = 1.

We next take up the case that G is the orientation preserving half of thegroup of motions of the tetrahedron, cube, or dodecahedron S. Let H be acyclic subgroup of maximal order in G, and let h be a geometric generator ofH. Then there is a face F of S kept invariant by h. Conjugate cp by a rotationof S2 so that cp(h) keeps the same face F invariant. As above, either P(h) = h,or cp(h) = h-1; in the latter case, compose cp with the isomorphism inducedby a half-turn interchanging the fixed points of h. Hence we can assume thatcp(h) = h.

Let j be a half-turn about an edge E on the face F. We now have to considerthe three cases separately.

If S is a tetrahedron, then cp(j) is also a half-turn about an edge of F. Rotateabout the center of F until cp(j) also keeps E invariant; then cp(j) = j.

If S is a cube, then cp(j) is a half-turn which keeps two opposite edgesinvariant. One of these edges lies on F, for if not, we would have cp(j o h o j) =cp(j)ohocp(j) = h-` = cp(h-1), while johoj is not a power of h. Hence we canrotate by a power of h so that cp(j) =j.

If S is a dodecahedron, then G contains 15 distinct half-turns. Five of thesehave fixed points on edges of F, five of them conjugate h into h', which, as weobserved above, cannot occur, and if cp(j) were one of the other five, then h o cp(j)would have order five, while h o j has order three.

In all three cases, we have shown that there is a rotation r of S2 so that r* o cpfixes both h and j; hence r* o cp is the identity on G.

V.I. Exercises 111

H.4. Proposition. Let q : G - ( be a type preserving isomorphism between Kleiniangroups, where G has exactly one limit point. Then N is a topological deformation,in particular, C also has exactly one limit point.

Proof. If G is parabolic cyclic, then so is C; we can normalize so that 0 is theidentity on G. If G is the infinite dihedral group, let J be the maximal paraboliccyclic subgroup of G. Then cp(J) = .7 is also a maximal parabolic cyclic subgroup;normalize G and t so that j(z) = z + I is a generator of both J and 7, and sothat cp(j) = j. Further normalize so that h(z) = - z is an element of G - J. Since(p(h) is a half-turn which conjugates j into its inverse, we can also furthernormalize so that cp(h) = h.

If G is a rank two parabolic group, then it suffices to assume that G is generatedby g(z) = z + 1, and by h(z) = z + i, and that l; is generated by g and h = rp(h),where h(z) = z + r, Im(r) # 0. Then, f(x + iy) = x + y Re(r) + iy Im(r) conju-gates g into itself, and h into h.

Observe that the map f, constructed above, commutes with z - -z; hencethe case that G has signature (0, 4; 2,2,2,2) is also taken care of.

The only cases left are the Euclidean triangle groups; we treat these together.In these cases, G has signature (0, 3; v1, v2, v3), where v, 5 v2 5 v3 < oo, and1/v1 + 1/v2 + I/v3 = 1. We choose generators g and h for G, where g has orderv3, h has order v,, and go h has order v2.

Let J be the maximal rank two parabolic subgroup of G, and let 7 = (p(J).We normalize G and CY so that J and 7 have their fixed points at co; then G andCY are both Euclidean groups. We normalize further so that g has its fixed pointsat 0 and oo, and h has its fixed points at 1 and oc. Similarly, we normalize C sothat g = rp(g) has its fixed points at 0 and oc, and h = cp(h) has its fixed pointsat 1 and oc.

Since cp is type-preserving, g = g±'. If necessary, we conjugate 6 by z - a, sothat g = g. If the signature of G is (0, 3; 2,3,6), or (0, 3; 2,4,4) then h is a half-turn,so h. Hence, in these cases, after appropriate normalization, 0 = G, and(P = 1.

The last case is that the signature of G is (0, 3; 3, 3, 3). In this case, an easycomputation shows that g o h is elliptic, while g o h-' is parabolic. Hence we musthave h = h. Again, after appropriate normalization, 0 = G, and p = 1. 0

V.I. Exercises

1.1. If G is a free Abelian Kleinian group of rank two, then every element of G isparabolic.

1.2. Let A and B be hyperbolic lines in H3 which meet at a point x. Let a and bbe the half-turns about A and B, respectively. G = <a, b> is discrete if and onlyif the angle between A and B is a rational multiple of it.


1.3. Let A and B be hyperbolic lines in 1-03 which do not intersect, even on thesphere at oc. Let a and b be the half-turns in A and B respectively, and let C bethe common perpendicular to A and B. Write the multiplier, or trace, of a o b interms of the (hyperbolic) distance between A and B, and the angle between them(measure the angle by parallel transport along Q. Hint: normalize so that C hasits endpoints at 0 and oo, and so that one endpoints of A is at 1; then the otherendpoint is at - 1.

I.4. Let P be a regular Euclidean polygon with v sides inscribed in the unit circle,and let Q be the suspension of P in lll', suspended from the points (0,0, 1) and(0, 0, - 1). The orientation preserving half of the group of motions of Q is thev-dihedral group (acting on §Z).

13. If G is a finite Abelian Kleinian group, then either G is cyclic, or IGI = 4.

1.6. Let G be the orientation preserving half of the group of motions of a regulartetrahedron. Then

(a) every element of G has order either two or three;(b) there is only one conjugacy class each of cyclic subgroups of orders two

and three;(c) IGI = 12;(d) G contains exactly one 2-dihedral group; every other proper subgroup is

cyclic.

1.7. Let G be the orientation preserving half of the group of motions of a cube.Then

(a) every element of G has order two, three, or four;(b) there is only one conjugacy class each of maximal cyclic subgroups of

orders two, three, and four;(c) IGI = 24;(d) G contains two conjugate subgroups of index 2; each is the orientation

preserving half of the group of motions of a tetrahedron;(e) G contains three conjugate 4-dihedral groups, four conjugate 3-dihedral

groups, and some 2-dihedral groups;(f) all subgroups of G not listed in (d) and (e) above are cyclic.

1.8. Let G be the orientation preserving half of the group of motions of adodecahedron. Then

(a) every element of G has order two, three, or five;(b) there is only one conjugacy class each of cyclic subgroups of orders two,

three, and five;(c) for v = 2, 3, 5, if a subgroup H of G contains all elements of order v, then

H=G;

V.I. Exercises 113

(d) G is simple;

(e)IGI=60;(f) G is isomorphic to the alternating group A5.

1.9. Let G be the Euclidean group with basic signature(a) (0, 3; 2,3,6), (b) (0, 3; 2,4,4), (c) (0, 3; 3,3,3), and let J be the maximal

parabolic subgroup of G. What is the index of J in G?

I.10. The group constructed in D.7 is a subgroup of the group constructed inD.6. What is its index?

1.11. Let j(z) = z + 1, and let w be a path in C from some point z to j"(z), m > 1.Then w fl j(w) # o.

1.12. Let C,, C,_., C", C,, be disjoint circles in C bounding a common regionD. Let g3 a M be such that g;(C) = C, and g,(D) fl D = 0. The group <g,...... , g"is called a classical Schottky group of rank n. If G is a non-elementary Kleiniangroup, then for every n, G contains a classical Schottky group of rank n. (Hint:Prove this for n = 2, then show that a classical Schottky group of rank 2 containsclassical Schottky groups of every rank.)

1.13. Let G be a Kleinian group with exactly two limit points, x and y. There isa simple G-invariant path W connecting x to y if and only if G is loxodromiccyclic, or G = Z2 * Z2.

1.14. There is a discrete subgroup G of 1_' where H'/G is not a 3-manifold.

1.15. Let G be a basic group, and let H c G also be a basic group. Then eitherH is cyclic, or H and G are of the same type (i.e., they are both finite, or bothEuclidean, or both Fuchsian).

1.16. If H is an Abelian subgroup of a Fuchsian group, then H is cyclic.

1.17. Let F be a Fuchsian group, and let x be a point on the circle at infinity.Then Stab(x) is either trivial, parabolic cyclic, or hyperbolic cyclic.

I.18. Let F be a Fuchsian group. Show that the convex region K (F) is F-invariant,and has no vertices (that is, if x e OK(F), then there is a hyperbolic line L, withxeL c aK(F)).

1.19. Let x and y be points on P, and let A be the hyperbolic line joining them.Let r e D_2 denote reflection in A, and let g e M be the half-turn with fixed pointsat x and y. Then r and g are both represented by the same matrix in PSL(2, Q.


1.20. Let A and B be hyperbolic lines in H2, and let a, b denote reflection in A,B, respectively. The element boa a L2 is hyperbolic if and only if A and B aredisjoint; it is parabolic if and only if they meet at exactly one point on the circleat infinity; it is elliptic if and only if they meet at exactly one point in 1-12.

1.21. Let H be a non-trivial normal subgroup of the Kleinian group G.(a) Show that if H is elementary, then so is G.(b) Show that if G is non-elementary, then A(G) = A(H).

V.J. Notes

B.4. This proof, including the idea of looking at ab-ba as an element of M, is dueto Jorgensen [37]. E.6. The proof given here is due to Jorgensen [36]. G.14. Aproof of this theorem can be found in Kra [43 pg. 68], or Beardon [11 pg. 254].H.I. A proof of the Nielsen isomorphism theorem can be found in Marden [54],or Zieschang, Vogt, and Coldeway [104]; see also Tukia [92].

Chapter VI. Geometrically Finite Groups

This chapter is an exploration of geometrically finite discrete subgroups of M;that is, groups that have (convex) fundamental polyhedra in H' with finitelymany sides. One of our main objectives is to give a criterion for a group to begeometrically finite in terms of its action at the limit set; this criterion will thenbe used in Chapter VII to show that, under suitable conditions, the combinationof two geometrically finite groups is again geometrically finite.

A geometrically finite group can also be described in terms of the underlyingmanifold (H' U Q(G))/G; this manifold need not be compact, but its non-compactends can be completely described in terms of a finite set of topologically distinctpossibilities.

VI.A. The Boundary at Infinity of a Fundamental Polyhedron

A.I. Let G be a discrete subgroup of L", and let D c H" be a fundamentalpolyhedron for G. By definition, D is convex, so the different sides of D lie ondifferent hyperplanes. In fact, different sides are identified by different elementsof G.

Proposition. Let sl 0 s2 be sides of D, and let gl and g2 he the corresponding sidepairing transformations; i.e., there are sides s; and sZ, so that Then

91 0 92.

Proof. The hyperplane on which si lies separates D from gl(D); in particular, itseparates s? from g1 (s2). Hence gI(s2) is not a side of D. Q

A.2. For a polyhedron D c B" we denote the relative boundary of D in B" byaD; we denote the intersection of the Euclidean boundary of D with SB" by 3D.The relative interior of aD in aH" is denoted by °aD.

If x e aD, and x lies on the boundary of the side s of D, then we say that sabuts x.

A.3. Thus far, the term "fundamental domain" is defined only for subgroups off fl (see II.G). The generalization to subgroups of V, acting on F"-', is obvious.

116 VI. Geometrically Finite Groups

Proposition. If D is a fundamental polyhedron for the discrete subgroup G of P",then °aD is a fundamental domain for G.

Proof. It is immediate from the definition that no two points of °aD are equivalentunder G. The sides of °aD are the boundaries of the sides of D. These are pairedby the same elements that pair the sides of D.

In 3", only finitely many supporting hyperplanes of sides of D meet anycompact set. If {sm} is a sequence of sides of D, and Q. is the supportinghyperplane of sm, then diaE(Qm) - 0; hence diaE(asm) - 0.

If x is a point on the boundary of °aD, where x does not lie on any side, thenthere is a sequence of sides of D accumulating to x; hence there is a sequence ofside pairing transformations { gm } so that for all z c- °aD, gm(z) - x. Hence x is alimit point of G. _

Let x be a point on the boundary of °aD, where xEQ. Then x lies on a sideof °OD, so there is a side s of D abutting x. There are only finitely many translatesof D in a neighborhood of s, hence there are only finitely many translates of IDin a neighborhood of x.

Finally, if z E Q (G), then choose a sequence of points {xm } in D3", with xm - z.For each xm there is an element gm e G, with xm E gm(D). There are only a finitenumber of distinct elements in this sequence, for otherwise, we would havediaE gm(D) -+ 0, contradicting the assumption that z E Q. Hence there is a g e Gwith g(z) in the closure of D.

A.4. In 3", a horosphere S is a Euclidean (n - 1)-sphere which is tangent to aB",and which, except for the point of tangency, lies in H". The open Euclidean ballbounded by S is called a horoball.

The point of tangency of S with aB" is the center, or vertex, of S. It is also thecenter, or vertex, of the horoball bounded by S.

A horosphere in H", centered at a finite point x, is a Euclidean sphere tangentto OH", which, except for the point of tangency, lies in H". A horosphere centeredat oo is a Euclidean plane parallel to aH".

A.5. Proposition. Let G he a discrete subgroup of ISO, where G containsj(z) = z + 1.Then the horoball T = { (z, t) E H 3 1 t > 1 } is precisely invariant under Stab(oo).

Proof. Let J = Stab(oo); by II.C.6, no element of J is loxodromic. Hence everyelement of J lies in A2, and every element of A2 keeps every horosphere centeredat oo invariant.

If

g = (c a d)

is any element of G, then by I1.C.5, either c = 0, in which case geJ, or Ici z 1.In the latter case, the radius of the isometric circle of g is at most one. Write

VI.A. The Boundary at Infinity of a Fundamental Polyhedron 117

g = q or, where r is reflection in the isometric circle of g, and q is a Euclideanmotion. Extend the actions of g, q, and r to H3. Then r is reflection in thehyperbolic plane whose boundary is the isometric circle of g. Since this isometriccircle has radius at most one, r(T) fl T = 0. The extension of q is a Euclideanmotion of the form q(x, , x2, t) t); hence q(T) = T, and q o r(T) fl T = 0.

A.6. An easy modification of the above yields the following.

Proposition. Let G be a Fuchsian group acting on H2, where G contains j(z) _z + 1. Then the horoball {zIIm(z) > I} is precisely invariant under Stab(oc).

A.7. If G is a Fuchsian group acting on 0-02 and G contains a parabolic element,then we can normalize G so that j(z) = z + 1 generates a maximal parabolicsubgroup. Let T be the horoball {z I Im(z) > 1). Since G is Fuchsian, Stab T =<j>. The exponential, f(z) = e2" `-- maps T onto a punctured disc, and is a covermap; that is, two points z and z' of T are equivalent under <j> if and only iff(z) = f(z'). We have shown the following.

Proposition. Let G be a Fuchsian group, acting on H2, where G contains theprimitive parabolic element j. Then there is a punctured disc, conformally embeddedin H2/G, so that under the natural homomorphism of n, ((0-02 fl °S2(G))/G) onto G,j corresponds to a small loop about the puncture.

We can visualize the puncture by normalizing G as above, and constructingthe Dirichlet region D in H2 centefed at 2i. The region Q = {zIIRe(z)l < 1/2,Im(z) > 2} is necessarily contained in D, and the closure of Q projects onto thepunctured disc.

A.8. For Kleinian groups the situation is somewhat more complicated. We stillhave an invariant horoball Tin 0.0 3, and there are only finitely many topologicallydistinct possibilities for Stab(T) (see V.D). This, however, yields no informationabout the 2-dimensional action, where there may or may not be punctures onQ(G)/G corresponding to any given maximal rank I parabolic subgroup.

Let G be a Kleinian group, and let J be a rank 1 Euclidean subgroup of G.J, or the fixed point x of J, is cusped if there is an open circular disc B C C whichis a (J, G)-panel. B is called a cusped region for J, or for x. The fixed point of J iscalled the center of B.

Similarly, J, or x, is doubly cusped if there are two disjoint open circular discsB, and B2, so that B, U B2 is precisely invariant under J in G. In this case,B = B, U B2 is called a doubly cusped region.

The two precisely invariant discs in a doubly cusped region need not beseparately precisely invariant. Let J = G = <z - z + l,z - -z>. Then the pairof discs {zIIm(z) < -1} U {zIlm(z) > l } is precisely invariant under J in G, butneither disc is separately invariant.


A.9. If B is a cusped region for x, then there is an element h of Id0, with h(x) = oo,so that z - z + 1 generates the parabolic subgroup of hJh-'. Choose E =(z l I Re(z)I < 1/21 as a fundamental domain for hJh-', and let C = h-' (E) fl B. Cis called a cusp for J, or for x.

A.10. Proposition. Let D be a finite sided fundamental polyhedron for the dis-crete subgroup G of A, and let x be a point of 8D. Then either xe(1(G), or J =Stab(x) is a Euclidean subgroup of G. Further, if J has rank 1, then x is doublycusped.

Proof. If x e '3D, then clearly x e 0. If x is an interior point of a side s of °t3D,then there is a side s', and a side pairing transformation g with g(s) = s'. Since aneighborhood of x is completely filled up by D, g-' (D), and points of s, x e Sl.Since D has only finitely many sides, the only other possibility is that there areat least two sides of D abutting x. Note that the isolated points of aD also lie onthe boundary of at least two sides.

Normalize so that x = oo. Since D has only finitely many sides, there is ahoroball T, centered at oo, so that T meets only those sides of D abutting oo. Ofcourse, these sides need not be paired with each other; we cut and paste to obtaina new object which, while it is no longer a polyhedron, at least has its sides nearoo paired. The cutting and pasting is done inductively as follows. If the sidesabutting oo are not paired, then there is a side pairing transformation g1, whichmaps a side abutting some point x, # oo on aD to a side abutting oo. ReplaceC, = D n (T) by g, (CI). If this new object has a side abutting oo paired witha side abutting some point x2 = 92' (00) s oo, then set C2 = D fl g2' (T), andreplace C2 by g2(C2); etc. After a finite number of steps, we arrive at a set D' withthe following properties.

D' is a not necessarily convex "polyhedron" bounded by a finite number ofsides, where some of the sides of D' lie on horospheres, rather than hyperplanes;D' is precisely invariant under the identity in G; there is a horoball T', centeredat oc, that meets only those sides of D' abutting oo; the sides of D' abutting ooare paired with each other.

The sides of D' abutting oo, if there are any, are pairwise identified by elementsof G. Since 0.03/G is complete, these identifications are all Euclidean motions;i.e., they are all either parabolic or elliptic (see IV.I.6-7). If there are no sides ofD' abutting oo, then all of (D' is contained inside a sufficiently large Euclideanball; i.e., oo e Q (G). If there is exactly one pair of sides abutting oD (this includesthe possibility of one side paired with itself), and the side pairing transforma-tion j is elliptic, then the outside of a sufficiently large Euclidean ball aboutthe origin meets only those sides abutting oo, so co is an elliptic fixed pointin 12.

If there are more than two sides abutting oo, then either one of the identifica-tions is parabolic, or there are two identifications which are elliptic; in the lattercase, the commutator of these two elliptic elements is parabolic with fixed point


oo (I.D.4). Once we have established a parabolic element in J, there can be noloxodromic element in J (II.C.6). We have shown that for every x on aD, eitherx e Q(G), or Stab(x) is a Euclidean subgroup of positive rank.

We now assume that oc is a point of aD, and that Stab(oo) is a rank IEuclidean group. We also assume that G is normalized so that j(z) = z + 1generates the parabolic subgroup of G.

The sides of D' abutting oe are paired by elements of J, which is either cyclicor the (2, 2, oo)-triangle group. In either case, ifs and s' are sides of D', paired byj e J, then either s = s', or s is parallel to s'.

Let E be the "polyhedron" bounded by the sides of D' abutting cc; I isactually a finite union of convex polyhedra, where two such polyhedra areadjoined along a common side. Since all the sides abut oo, these_ polyhedra areconvex in both the Euclidean and hyperbolic sense. Let E = °8E; then E is aunion of a finite number of Euclidean convex polygons in F2, where any two ofthese convex polygons are either disjoint or have a common side. The sides of Eare paired by elements of J; as above, if s and s' are sides of E, and j e J pairs swith s', then either s = s', or s and s' are parallel.

Since J has rank 1, E has at least one pair of parallel sides on its boundary;these are identified by a generator j of J. If E has exactly one pair of parallel sides,then these sides cannot be parallel to the real axis; hence there is a number b > 0,so that B = {zII ImzI > b} intersects only that pair of sides. Then B is preciselyinvariant under J.

Consider Im(z) to be height. If there is no highest point in E, then, since Ehas only finitely many sides, at sufficiently large height, we see only one pair ofsides, necessarily identified by j. In this case, there is a cusped region of the form{zllm(z) > b}. Similarly, if there is no lowest point in E, then there is a cuspedregion of the form {zIIm(z) < -b}, b > 0.

Suppose there is a highest point z t . If z t is unique, then it is the fixed point of ahalf-turn. Since E only extends downwards from the highest points, it is easy tosee that the set of highest points must be discrete. For the same reason, it is alsoeasy to see that the sides abutting these points cannot all be identified bytranslations. We have shown that if E has a highest or lowest point, then J is notcyclic. We now assume that E has a highest point, and that the fixed points ofthe half-turns in J all have height zero.

Now cut and paste once more to obtain a new "polygon" E', where E' isentirely contained in the lower half-plane. It is clear that except for sides lyingon the real axis, the side pairing transformations of E' are all translations. Sincethere are sides of E' extending into the lower half-plane, we can conclude asabove that E' has no lowest point.

A.11. Proposition. Let G be a discrete subgroup of fyl, and let Jt and J2 benon-conjugate maximal Euclidean subgroups of G, where Jam, has its fixed point at

Then there are horoballs Tt and T2, where T. is centered at x. so that (T1, T2)is precisely invariant under (Jt,J2) in G (see 11.1.6).


Proof. Normalize so that x, = oo, x2 = 0, and so that j,(z) = z + I is an elementof J, . Start with the horoballs T, = { (z, 01 t > 1 } and T2, where T2 is a disjointfrom T,, and precisely invariant under J2. Observe that the only elements of NAwhich conjugate J, onto itself are other Euclidean translations; hence any suchelement also stabilizes T,. Suppose there is a sequence {gm} of elements of G,with having non-trivial intersection with T,,m = {(z,t)It > m}. Let pmbe the Euclidean radius of and let zm = gm(0) be its center. Let am(z) =

where A. is chosen so that am(T2) is a horosphere of (Euclidean) radiuspm. Let bm be the translation z -a z + zm. Since bm o am(T2.) and gm(T2) are horo-spheres of the same size and vertex, there is a (Euclidean) rotation cm, with fixedpoints at zm and oo, so that gm = Cm o bm o am. Since Am' oo, for any j2 a J2, thelower left hand entry of am oj2 o a.' tends to 0. The absolute value of thislower left hand entry (the reciprocal of the radius of the isometric circle) is leftunchanged by conjugation by a Euclidean motion. Hence form sufficiently large,the radius of the isometric circle of gm 0j2 o g.' is greater than 1; this contradictsII.C.5.

A.12. One can easily extend the statement and proof above from two Euclideansubgroups to a finite set {J...... J.) of nonconjugate maximal Euclidean sub-groups. We obtain a set of horoballs {T,..... T,,} so that (T,,..., T,,) is preciselyinvariant under (J1..... J,,) in G.

A.13. Proposition. Let D he a finite sided fundamental polyhedron for the discretesubgroup G of M, and let x e aD be the fixed point of a rank 2 Euclidean subgroupof G. Then there is a precisely invariant horoball T, centered at x, so that D - Tis bounded away from x.

Proof. Let T be any precisely invariant horoball centered at x. Normalize so thatx = oo. Then a sequence of points {(zm,tm)} of D - T can approach oc only iftm is bounded. Since ('T/{Stab(oo)} has finite area, this cannot happen.

A.M. The result above is not in general true if J = Stab(x) has rank 1, but thereis an obvious modification. By A.10, there is a precisely invariant doubly cuspedregion B = B, U B2. Let H. be the half space in 0.03 whose boundary is Bm, andlet T' = T U H, U H2. T' is a (J, G)-panel (see VI.D for a description of T'/J). Theset T' is called an extended horoball centered at x. If we normalize so thatz - z + I generates the parabolic subgroup of J, then we can take T' to a set ofthe form: { (z, t) e H 3 t > 1) U { (z, t) e HI I I Im(z)I > a}, for some a > 0.

Proposition. Let D he a finite sided fundamental polyhedron for the Kleinian groupG, and let x e aD, where Stab(x) is a rank I Euclidean subgroup of G. Then thereis a precisely invariant extended horoball T', centered at x, so that D - T' isbounded away from x.


Proof. Normalize so that x = oo, and so thatj(z) = z + 1 generates the parabolicsubgroup of Stab(oo). Since D is convex and has finitely many sides, all the sidesof D not abutting oo are contained in a large Euclidean ball. We construct theset E, as in A.10, to be bounded by the traces of those sides of D abutting ec,together with the appropriate translates of those sides abutting the points on 8Dequivalent to oo. These sides extend to co only in an infinite strip, or half-infinitestrip, that is not parallel to the real axis. Hence outside of a large Euclidean ball,all points of D either have large height or large imaginary part.

A.M. Suppose x1, ..., x are points of 8D, and T. is a horoball or extendedhoroball at xm. We say that {T1,..., is precisely invariant relative to D if eachT. is precisely invariant under Stab(xm), and, whenever there is a g e G withg(xj) = x,t, then g(Tj) = T.

We say that the fundamental polyhedron D for G is essentially finite if thereis a finite set of horoballs or extended horoballs where these areprecisely invariant relative to D, and D - U T. is bounded away from A(G).

Proposition. Let D be a finite sided fundamental polyhedron for the discretesubgroup G of L3. Then D is essentially finite.

Proof. We already know that we can find a T for each x; so that D - U T. isbounded away from A(G); this follows from A.10, 13-14. We also know that wecan find disjoint precisely invariant horoballs, Ti, ..., T., which are eitherpairwise equivalent, or pairwise precisely invariant, according as the centers areequivalent or inequivalent; this is the content of A.11. It remains to prove thatif x ..., x,, are inequivalent doubly cusped parabolic fixed points on dD, thenthere are precisely invariant doubly cusped regions B. for xm, so that (B,,...,is precisely invariant under in G; for then the corre-sponding sets of hyperbolic half-spaces will also be precisely invariant.

We assume without loss of generality that n = 2, and we assume that thereis a sequence {zm} of points lying in a cusp C, c B1, with zm -4 x,, and there isa sequence of elements { gm } of G, where gm(zm) converges to x2 inside B2. Choosea cusp C2 in B2, and extend it to a fundamental domain CZ for J2 = Stab(x2).For each in, replace gm by an element of the form j o gm, j e J2, so that gm(x,) E C.

Since gm(x,) a B2, and x2 is doubly cusped, the points gm(xl) are boundedaway from x2. Then, since the sets {gm(B,)} are either equal or disjoint, theirspherical diameter tends to zero; hence we can choose B2 smaller, so that it doesnot intersect any of these sets. Q

A.M. A minor modification of the above proof yields some information aboutgroups that need not be geometrically finite.

Proposition. Let x1 and x2 be non-conjugate cusped rank I parabolic fixed pointsof the Kleinian group G. Then there are open circular discs, B1 and B2, so that(B1, B2) is precisely invariant under (Stab(x, ), Stab(x2)) in G.


VI.B. Points of Approximation

B.1. Let G be a discrete subgroup of P". A point x in S" is a point ofapproximation for G if there is a sequence {gm} of distinct elements of G so thatIgm(x) - g.(z)I Z b > 0 uniformly on compact subsets of S"-' - {x}; note thatfor n = 2, this agrees with the definition given in II.D.4.

A point x e a W" is a point of approximation for the discrete subgroup G ofL if j(x) is a point of approximation for jGj-', where j e L"+' maps W onto B".

8.2. If G is Kleinian, then given the sequence {gm}, and given any pointy, the setof points for which gm(z) -+ y is both open and closed in Q; hence a point ofapproximation for a Kleinian group is necessarily a limit point (see B.6).

If G is elementary with no limit points, then obviously A(G) contains no pointsof approximation.

It is easy to see that if G is an elementary Kleinian group with one limit point,then the one limit point is not a point of approximation (see B.8).

If G is an elementary Kleinian group with two limit points, then they are bothloxodromic fixed points; these are points of approximation.

B3. Proposition. A point x e S is a point of approximation if and only if thereis a point z E B", and there is a sequence { gm } of distinct elements of G so thatlgm(x) - gm(z)I z S > 0.

Proof. Assume first that x is a point of approximation and let {gm} be thesequence of elements given by the definition. Let y, and y2 be points of S""'different from x, and let A be the geodesic connecting y, to y2. Choose asubsequence { gm } so that gm(x) - x', gm(y,) - y', s x', and gm(y2) -' y2 # x'. Letz be some point on A. Then either gm(z) -' y, or gm(z) - y2.

Now assume that there is a ze B" so that Igm(z) - gm(x)I >- 8 > 0. Choose asubsequence so that gm(z) converges to some point y. Then by IV.G.9, gm(w)converges to y uniformly in compact subsets of the complement (in t") of somepoint y'. Since Igm(z) - gm(x)l >- S, we must have y' = x. It follows that x is apoint of approximation. p

B.4. The proof above yields a bit more.

Proposition. A point x e S"-' is a point of approximation if and only if there is asequence { gm } of distinct elements of G, so that gm(x) -+ x', and gm(z) - z' # x',for allzE 3".

BS. Proposition. Let D be a fundamental polyhedron for the discrete subgroup Gof P", and let x be a point of D. Then x is not a point of approximation.

VI.B. Points of Approximation 123

Proof. Let {gm} be any sequence of distinct elements of G with gm(x) -4 x'. Let Lbe a semi-infinite hyperbolic line segment lying entirely inside D, where L hasone endpoint at x. Since diaE gm(D) - 0, diaE gm(L) -+ 0; hence gm(z) - x' for allzonL.

B.6. Proposition. Every point of approximation is a limit point.

Proof. Combine B.5 with A.3.

M. Proposition. Let x be a parabolic fixed point of the discrete subgroup G of M.Then there is a fundamental polyhedron D for G, with x e 9D.

Proof. We can assume that the parabolic fixed point is at oc; let T be a preciselyinvariant horoball. Let xo be some point of T. and let D be the Dirichlet regionfor G centered at xo. Every translate of xo by an element jeStab(oo) preservesthe height of xo, so the perpendicular bisector of the line between xo and j(xo)abuts oo. If g e G - Stab(oo), then g(xo) is lower than xo, so if y is any point onthe line from xo to oo, then y is closer to xa than it is to g(xo). It follows that theline from xa to oo lies in D, hence oo e OD.

B.8. Corollary. Let x be a parabolic fixed point of the discrete subgroup G of M.Then x is not a point of approximation.

M. Proposition. Let G be a Kleinian group, normalized so that oo c -'Q. The pointx is a point of approximation if and only if there is a sequence {gm} of distinctelements of G, where the isometric circle 1m of gm has radius pm and center am, andthere is a constant K > 0, so that for all m, Ix - aml 5

Proof. First assume that x is a point of approximation. Then there is a sequence{gm} of distinct elements of G so that Igm(oo) - 9m(x)I >: b > 0. Set hm = g-', andset ym = gm(x). Choose a subsequence so that the radius pm of the isometric circle1m of hm is less than 6. Let pm denote reflection in the isometric circle of hm. Sinceh;'(oo) is the center of Im, and Igm(x) - gm(0o)I = lym - h'(oo)l > b, it followsthat l pm(ym) - h,M'(cc)l S pm/8. The result now follows from:

IPm(Ym) - h;'(cc)I = Ihm(ym) - hm(co)I

= Ix - go"'(00)1

Next assume that we are given the sequence {gm}. Let pm denote reflec-tion in the isometric circle of gm. Since Ix - aml S Kpm, Ipm(x) - aml =Igm(x) - gm(oo)I > K°'. Choose a subsequence so that gm(oo) -+ y; then gm(z) -y, for all z 9A x. Hence, for m sufficiently large, and for all z in a compact subsetof't - {x}, there is a constant K' so that Igm(z) - gm(x)l >: K'.


VI.C. Action near the Limit Set

U. Proposition. Let D be an essentially finite fundamental polyhedron for thediscrete subgroup G of M, where oo is a parabolic fixed point on jD, and let {z.)be a sequence of points of C which are G-translates of some point y. Then thereare elements { jm} in J = Stab(oo), with jm(Zm) contained in a bounded subset of C.

Proof. If J has rank 2, the result is obvious. If J has rank 1, then let T be a preciselyinvariant extended horoball centered at oo, and let B = aT. Modulo J, there isat most one translate of y in B; hence we can assume that all the zm lie outsideB, where the result is easy. El

C.2. Proposition. Let D he an essentially finite fundamental polyhedron for thediscrete subgroup G of M. Then every limit point of G, which is not a translate ofa point of 8D, is a point of approximation.

Proof. Let x be such a limit point of G, and let L be a hyperbolic line passingthrough D with one endpoint at x. Let y be the other endpoint of L. We canassume without loss of generality both that y is not a parabolic fixed point, andthat L does not lie in any translate of a side of D. It is clear that L cannot lie inany one translate of D, hence there is a linearly ordered sequence of points {xm}on L with xm -+ x, and there is a sequence of distinct elements {gm} in G, whereZm = gm(xm) lies in D. Choose a subsequence so that zm - z', gm(x) -' x', andg ,.(Y) - Y'.

The sequence of lines L. = gm(L) either converges to a line, or it convergesto a point. If {Lm} converges to the line M, then the endpoints of M are x' andy' # x', and z' is some point of M. Otherwise, {Lm} converges to the pointx'=yz.

Suppose first that z' is an interior point of M. On Lm, zm = gm(xm) separatesgm(x) from gm(xt ). Since z' is an interior point of B', Igm(x) - gm(xt )I is boundedaway from 0.

Similarly, if z' = y' # x', then gm(x t) -- y', so x is a point of approximation.If z' = x', then since every zm lies in D, x' c- ijD; hence x' is a parabolic fixed

point. Let T be a precisely invariant horoball or extended horoball at x'. SinceD - T is bounded away from x', the lines L. all pass through T If these linesconverge to the line M, then M also passes through T.

At this point, we pick up the case that the lines L. converge to the point x'.Since y is not a parabolic fixed point, x' is not a translate of y; hence there is asequence of elements { jm } of J = Stab(x') wherejm o gm(y) is bounded away fromx'. We can in fact assume that these points converge to some point, call it y'.Since each L. passes through T, so does each jm(Li).

In any case, under the assumption that z' = x', we have constructed asequence of lines {hm(L)}, with hm =jmogm (if the sequence of lines {Lm} con-verges to a line, then every jm = 1), so that each hm(L) passes through T;hm(x) -+ x', the center of T; and hm(y) - y' 96 x'.

VI.C. Action near the Limit Set 125

Consider the points {hm(z,)}. If all these points lie outside T, thenIhm(x) - hm(z, )) is bounded away from zero. If they all lie inside T, then they areall J-equivalent. Then for every in, there is a km E J, with km o hm(z,) = h, (z,). Sincez, is equivalent to a point of D, it is fixed only by the identity, so km o hm = h'.This says that gm = Jm' o k,' ofi o gi =.Im o gi Since the gm are all distinct, so arethe].. The sequence of lines { jm o g, (L)} converges to the line M; since the jm aredistinct elements of the Euclidean group J, this can occur only if one of theendpoints of g, (L) lies at the fixed point x'. Since y is not a parabolic fixed point,and x is not equivalent to any point of 3D, we have reached a contradiction; weconclude that the points hm(z,) all lie outside T, so that I hm(z,) - hm(x)l >_ b > 0.

O

C-3. Observe that the proof above is not restricted to Kleinian groups, but appliesequally well to groups of the first kind; that is, discrete subgroups of M that donot act discontinuously anywhere on t.

Corollary. Let G be a discrete subgroup of fa+0 with an essentially finite fundamentalpolyhedron D. Then every limit point of G is either a doubly cusped rank I parabolicfixed point, or is a rank 2 parabolic fixed point, or is a point of approximation.

C.4. Theorem. Let D be a (convex) fundamental polyhedron for the discrete sub-group G of U. If every limit point of G is either a doubly cusped rank I parabolicfixed point, or a rank 2 parabolic fixed point, or a point of approximation, then Dhas finitely many sides.

Proof. Assume that D has infintely many sides. Then there is a sequence of sides{sm } that accumulates at some point x of 3D. Let Q. be the hyperplane on whichsm lies; by definition of a polyhedron in B3, diaE(Qr) -+ 0. Since there is a trans-late of D on either side of Qm, x is a limit point of G. By B.5, x is not a point ofapproximation. Hence x is a parabolic fixed point.

We normalize so that x = oo, and so that J = Stab(oe) containsj(z) = z + Ias a primitive element.

Since D is convex, the Euclidean closure of D is also hyperbolically convex;in particular, if (z, t) is some point of D other than oo, then the hyperbolic line be-tween (z, t) and oo is contained in D.

Let A be the set of points z e C for which there is a t > 0 so that (z, t) a D. Ifz, and z2 are points of A, then for all sufficiently large t, the points (z,, t) and(z2, t) all lie in D; hence z, and z2 cannot be equivalent under J. We concludethat A is precisely invariant under the identity in J.

Let z, and z2 be points of A. Then there are numbers t, and t2 so that (z t, )and (z2, t2) are both in D. Let L be the hyperbolic line segment joining thesepoints. Since D is convex, L is in D. Write the points of L as (z,,, ta) and observethat the points {z,} lie on the Euclidean line segment joining z, to z2. We haveshown that A is a convex subset of C.

126 Vl. Geometrically Finite Groups

CS. The sequence of sides {sm} converges to co. By passing to an appropriatesubsequence, it suffices to assume either that none of these sides abuts oo, orthat they all do. We take up the first case first, that is, we assume that oo doesnot lie on the boundary of any Qm.

Pick some point (zm, tm) on each sm so that (zm, tm) - oo; since sm does notabut oo, zm e A. There is a relatively open set of points we could choose on eachsm, hence we can assume that the points zm do not all lie on a line. If J has rank2, then since A is precisely invariant under the identity in J, A has finite area;also A has non-empty interior, so A is bounded. In this case, each zm is bounded,and t,,, - oo. The only way this can occur is for the supporting hyperplanes Q. toapproach the vertical; i.e., the Qm all pass through a compact subset of H'; thiscannot occur.

We next take up the case that J has rank 1. Normalize further so that anyhalf-turn in J has its finite fixed point on the real axis, and so that A is containedin the strip between some line L and L' = L + 1, where L is not parallel to thereal axis. For each sm there is a corresponding side s;,,, and there is a side pairingtransformation gm, with gm(sm) = s..

Assume that tm is bounded, and look at the points zm in A. If Im(zm) is alsobounded, then since L is not parallel to the real axis, Re(zm) is bounded;this is impossible, for (zm, tm) - oo. We conclude that either tm - oo, or IIm(zm)I-oo.

Consider the semi-infinite line segment M. from gm(zm, tm) to oo; note thatMm c D. One endpoint of g,' (Mm) is at (zm, tm), and the other at g;' (oo), whichhas bounded imaginary part. There is an element jm e J so that the real part ofjm o g,,' (oo) is also bounded. An element of J is either a purely real translation,or a Euclidean half-turn about a real point; hencejm leaves the absolute value ofimaginary part and the height of (zm, tm) unchanged. Then jm o g,' (Mm) has oneendpoint in a bounded part of C, and the other endpoint has either unboundedimaginary part or unbounded height. In any case, the lines jm o g;' (Mm) all passthrough a compact subset of H3. Since M. c D, and the translates of D form alocally finite tesselation of H3, there are only finitely many distinct elements ofthe form jm o g,"

Note that if jm o gin' = jk o gk 1, then gm(oo) = gk(oo). Hence we can assume thatthere is a subsequence { gm } where gm = g, o jm (we can assume that j, = 1). Sinces, does not abut oo, the point g,(co) = gm(oo)0 D. Now look at the line N. from(zm, tm) to oo, and consider gm(N,). Choose a subsequence so that gm(zm, tm) - Y.Since gm(zm, tm) a OD, y e aD. We have shown that the endpoints of gm(Nm) con-verge to different points; hence gm(NN) converges to some line N. By A.1, theelements gm are all distinct; hence the tesselation of I-0' by translates of D is notlocally finite near N. We have shown that there is no sequence of distinct sides{sm}, where no sm abuts oo, and sm - oo.

C.6. We next take up the case that there is a sequence of sides {sm}, where eachsm abuts oo. For each m, there is a corresponding side s;,,, and a side pairingtransformation gm, with gm(sm) = s;.

VI.C. Action near the Limit Set 127

The points gm(oo) all lie in M. If there were infinitely many distinct suchpoints, there would be a subsequence accumulating to some point z e BD.Obviously z e A; hence z is a parabolic fixed point. We temporarily renormalizeand place z at eo, so that there is a sequence of limit points zm in aD with zm oo.

The points zm are all equivalent under G, they all lie in A, and they all lie inthe complement of the boundary of the possibly extended horoball at co. Weobserved above that such a set is necessarily bounded. We have shown that thereare only finitely many distinct points in the set {gm(oo)}, where gm pairs a sideabutting oo with some other side.

We already know that there cannot be infinitely many sides that accumulateto oo, but do not abut oo. Therefore, for to sufficiently large, the horoballTo = {(z,t)It > to} meets only those sides of D abutting co.

If J has rank 2, then the Euclidean area of BTo/J is finite; hence D fl BT0is bounded; hence A, which is contained in the vertical projection of D fl BTo, isalso bounded. If there were infinitely many sides abutting oo, they would all passthrough a bounded part of BTo, which is impossible.

If J has rank 1, and there are no points of OD equivalent to oo, other than 00itself, then the sides sm all project to lines on BTo, which is convex and preciselyinvariant under J in G. Since J has rank 1, there are at most a finite number ofelements of J that can identify the sides of D fl BTo. Since distinct sides of D areidentified by distinct elements of G, D has only finitely many sides abutting 00.

Finally, we take up the case that J has rank 1, and there are a finite numberof points equivalent to oo on aD. Exactly as in A.10, we construct a new set D',which is no longer a polyhedron, since some of its "sides" lie on horospheres,rather than hyperplanes, where D' is precisely invariant under the identity in G.The sides of D' are paired by elements of G, and while these elements need notall be distinct, there are at most finitely many of them equal to any given one (wecut and paste only finitely many times; for each such operation, the new sidepairing transformations are obtained from the old ones by conjugation by a fixedelement of G).

There do not exist infinitely many sides of D that accumulate to 00, withoutabutting oo; similarly, there do not exist infinitely many sides accumulating to apoint x e aD, equivalent to oo, without abutting x. Hence there is a horosphereT, = {(z, t)I t = t, }, so that every side of D' that intersects T, abuts oo. Clearly,T, is precisely invariant under J in G; also, 6 = T, fl D' is precisely invariantunder the identity in J.

Let B be the vertical projection of ,6 to C; that is, 1 3 = {z l(z, t,) a 6}. Since Jis a Euclidean group, 13 is also precisely invariant under the identity in J.

6 is a connected finite union of (Euclidean) convex polygons, but need not beconvex itself. The sides of 6 are paired by elements of J. Each of the convex poly-gons, which make up 6, is either bounded, in which case it has finitely many sides,or is contained in a strip between two parallel lines, where the horizontal distancebetween these lines is at most 1. For each such strip, there can be at most finitelymany distinct sides that are paired with sides of the same strip, for if there wereinfinitely many, infintely many of them would be paired by the same element of J.


The only possibility left to be investigated is that there are two of these strips,call them S and S', where (i) S and S' each contain infinitely many sides of D, (ii)there is a sequence {sm} of sides of S, where the paired side, s,;, lies in S', and (iii)the side pairing transformations jm, with jm(sm) = s;,,, are all distinct.

Since D fl s is convex, the sequence of sides {sm} has a limiting direction;similarly, the sequence of sides {s;,,} approaches some limiting direction. Theselimiting directions are those of the parallel lines bounding the respective strips.For each m, the sides sm and s, are identified by an element of J, and so areparallel. Hence the limiting directions are parallel. There are at most a finitenumber of elements of J that can identify a point of a (non-horizontal) infintestrip of bounded width with a point of a parallel strip, which is also of boundedwidth.

C.7. We combine the results of A.15, C.3, and C.4, to obtain three equivalentdefinitions of a geometrically finite Kleinian group.

Theorem. Let G be a discrete subgroup of ICU, and let D be a (convex) fundamentalpolyhedron for G. The following statements are equivalent.

(i) D has finitely many sides.(ii) D is essentially finite.(iii) Every limit point of G is either a rank 2 parabolic fixed point, a doubly

cusped rank I parabolic fixed point, or a point of approximation.

VI.D. Essentially Compact 3-Manifolds

D.I. Let G be a discrete subgroup of Al,Aand let M = H3/G. If G is torsionfree, M is called the associated 3-manifold, or sometimes simply the associatedmanifold. An element of finite order has a line of fixed points in H3; this lineprojects to a curve on M where the hyperbolic metric is singular. In this case,M, with its singular hyperbolic structure, is called the associated orbifold. If weignore the hyperbolic structure, then M is always a manifold; see V.E.1 1.

D.2. We observed in C.7 that G is geometrically finite if and only if there is anessentially finite fundamental polyhedron D for G. In this section we explore themeaning of this condition in terms of the associated orbifold M = (H3 U Q(G))/G.In some sense, M is essentially compact; that is, there is a natural compactifica-tion of M as a manifold. Our goal here is to exhibit this natural compactification.

D.3. The statement that D is essentially finite means that there is a finite set ofdisjoint suborbifolds, S,,..., S of M, where each of these is the image under theprojection of a precisely invariant horoball or extended horoball, so that if weremove U S. from M, the resulting orbifold is compact.

VI.D. Essentially Compact 3-Manifolds 129

Fig. VI.D.1

Topologically, we know all the possibilities for the sets Sm. Let x be a pointon 8D, let J = Stab(x), and let T be the precisely invariant horoball or extendedhoroball centered at x. If J has rank two, then T is a horoball, and T/J is of theform U x (1, oo ), where U is a surface of signature (1, 0), (0, 4; 2,2,2,2),(0,3; 2,3,6),(0, 3; 3, 3, 3), or (0, 3; 2,4,4). In this picture, 8T/J corresponds to U x { I).

We could topologically compactify M near this end by adding in a copy ofU to correspond to U x { oo }; there is also another natural compactification thatcloses up the end.

If U has signature (1,0), then we regard U x (1, oo) as an open solid toruswith its central core loop removed (see Fig. VI.D.1). We can also regard the solidtorus as being the cylinder {x2 + y2 < 1) in (x, y, t)-space, factored by the group{t - t + n, n e 7L}. We identify U x [ 1, oo) with this solid torus, where the imageof the line x = y = 0 has been removed. We compactify M by re-inserting thiscircle. Similarly, if U has genus 0, then we can regard U x (1, oo) as the 3-ballwith its center removed; we compactify M by re-inserting the removed point.

While the above compactification is in some sense more natural than that ofadding in a copy of 8T/J at oo, it does change either nl or n2.

D.4. If J has rank one, then we normalize so that x = oo, and so that j(z) = z + 1generates the parabolic subgroup of J. Then T is an extended horoball of theform T = J (z, t)l t > 1) U J (z, t) l I lm (z) I > a), for some a > 0. Let b = max(a, 1),and set T' = {(z, t)I t2 + (Im(z))2 z b2 }. We can regard T' as being T with itscorners rounded. Since 1' c T, T' is also precisely invariant under J in G, andD - T' is bounded away from oo.

We regard T' as a product of three factors: the interval [-n, n], the interval[b, co), and the real line. The endpoints of the First interval correspond to theboundary of T' in 80-03.

D.S. If J is cyclic, then every element of J keeps every point of the first twointervals fixed, and J acts on the third interval by translation. Thus T'/J is theproduct [-a, n] x [b, oo) x S'. We can regard the product of the last twofactors as a (closed) punctured disc (i.e., a set of the form D' = {z10 < I z 1 S 1}),so that T'/J is the product: D' x [ - it, n]. The two punctured discs D' x {n} andD' x ( - n) on the boundary of T'/J are exactly the punctured discs B/J on DIG,


am am

Fig VI.D.2

Fig VI.D.3

where B is the precisely invariant doubly cusped region {zIIIm(z)I z b}. Wecompactify M near the projection of T' by adding in a curve C to correspond tothe product of the puncture and the interval. Note that the endpoints of C arespecial points of order oo on 8M (see Fig. VI.D.2).

D.6. If J is the (2, 2, oo)-triangle group, then we further normalize so that J isgenerated by j and g(z) = - z. We now write T' as the product of the same factors,but in a different order; T' is the product of the interval [b, oo), the real line, andthe interval [ - n, n]. Then g acts on the third interval as the involution x - - x;every element of J keeps the first interval pointwise fixed; the action of J on thereal line is the usual action of its parabolic cyclic subgroup on the real line. Aswe saw above, the action of J on the product of the first two intervals yields a(closed) punctured disc D'.

Let W be the product of the punctured disc D', with the closed interval[ -1t, n] given above. The action of g on T' projects to a homeomorphism(p: W - W, where (p(z, t) = (z, - t). In this case, T'/J is W/sp. We can realize W/cpas D' x [ - x, 0], where D' x { - n} lies on 8M, and D' x {0} is a punctured discin the interior of M, with two lines of ramification R and R' (see Fig. VI.D.3).In this picture, D' x {0} is the image of T' fl {(z, t)I Im(z) =0). The plane{Im(z) = 0) contains the axes of all the half-turns in J; these project onto theramification lines R and R'. Since all the axes in J extend to oo, R and R' extendto the puncture in D' x (0). The ramification lines R and R' extend from thepuncture in D' x {0} out past its boundary. They do not reenter T'/J, but wehave no other knowledge of their extent on M. We can again compactify M byfilling in a missing curve C; in this -case, C is the product of the puncture in D'with the interval [ - n, 0].

V1.E. Applications 131

VI.E. Applications

E.I. We start with an application of A.3.

Theorem. A geometrically finite Kleinian group is analytically finite.

Proof. Let D be a fundamental polyhedron for the geometrically finite Kleiniangroup G._By A.3, °dD is a fundamental domain for the action of G on 0, and ofcourse, °aD has only finitely many sides. Since °3D has finitely many connectedcomponents, so does S = Q/G. Each side contains at most one elliptic fixed point,necessarily a fixed point of a half-turn; except for this one possibility, all theinterior points of sides lie in °Q. Since there are only finitely many sides andvertices, there are at most finitely many special points (with finite ramificationnumber) on S. Each vertex is either a point of °Q, or an elliptic fixed point, or acusped parabolic fixed point (see A. 10). The projection of a cusped region is apunctured disc; hence there is a compact (disconnected) Riemann surface S.containing S, so that S - S is a finite set of points.

E.2. Condition (iii) of C.7 is independent of the choice of fundamental polyhedron.

Theorem. Let G be a discrete subgroup of M. If any one (convex) fundamentalpolyhedron for G has finitely many sides, then every (convex) fundamental poly-hedron for G has finitely many sides.

E.3. Proposition. Let G be a geometrically finite subgroup of M, let D be any(convex) fundamental polyhedron for G, and let x be a parabolic fixed point of G.Then there is an element g e G, with g(x) e D.

Proof. By C.2, every point of A which is not a translate of a point of dD is a pointof approximation, and by B.8, x is not a point of approximation.

E.4. Let G be a Fuchsian group acting on B2, and let D be a fundamental polygonfor G. Each side of D determines a hyperbolic plane in H3; the pairing of thesides of D is also a pairing of these hyperbolic planes. It is easy to see that theseplanes bound a fundamental polyhedron for G as a Kleinian group. Combiningthe above with V.G.14, we obtain the following.

Proposition. A Fuchsian group is finitely generated if and only if, qua Kleiniangroup, it is geometrically finite.

E.S. We will need the following lemma for the next two applications.

Lemma. Let J be a maximal rank 1 Euclidean subgroup of the Kleinian group G.Suppose there is a circular disc B, where B is J-invariant, and B fl 4(G) = 0. ThenB contains a cusped region for J. Similarly, if there are two disjoint open circular


discs, B, and B2, where Bt U B2 is J-invariant, and A(G) fl (B, U B2) = 0, then Bcontains a doubly cusped region for J.

Proof. The proof in the two cases is essentially the same; we prove only the secondstatement. Normalize so that j(z) = z + 1 is a primitive element of J; in partic-ular, J = Stab(co). There is a number b' > 0, so that B' = {zIIIm(z)I > b'}contains no limit points of G. Let B = {zIIIm(z)l > b}, where b = b' + 1. Let gbe any element of G - J. The isometric circle I of g has radius at most one, andits center, being a limit point, lies outside B'. Hence I lies entirely outside B.Similarly, the isometric circle I' of g-t also lies outside B. Since B lies outside I,g(B) lies inside I', which is disjoint from B.

E.6. Proposition. Let G be a discrete subgroup of M, where G contains a geo-metrically finite subgroup H of finite index. Then G is geometrically finite.

Proof. Since [G : H] < co, A(G) = A(H). It is immediate that every point ofapproximation for H is also a point of approximation for G. If J is a Euclideansubgroup of G, then J fl H is a Euclidean subgroup of H, of the same rank. If Jis doubly cusped, as a subgroup of H, then since A(G) = A(H), it is also doublycusped as a subgroup of G.

E.7. Proposition. Let ( be a deformation of the geometrically finite Kleinian groupG. Then t is geometrically finite.

Proof. Let f : C - C be the deformation that conjugates G into 6, and letgyp: G -+ C be the induced isomorphism. Since qp is an isomorphism, it preservesrank 2 Euclidean subgroups. Also, if J is a doubly cusped rank 1 Euclideansubgroup of G, then there is a doubly cusped region B = B, U B2 for J. Whilef(Bm) need not be circular, it is invariant under the Euclidean subgroup qO(J) in6. It is easy to see that f(Bm) contains a cp(J}invariant circular disc. Sincef(A(G)) = A(C), (p(J) is a doubly cusped rank I Euclidean group.

If x is a point of approximation for G, then there is a sequence of distinct ele-ments { gm} with gm(x) -+ y, and gm(z) -+ y' 0 y, for all z # x. Then (p(gm) of(x) =f o gm(x) --+ fly), and for all z # x, gp(gm) of(z) = f o gm(z) -+ fly') # fly). Hencef(x) is also a point of approximation.

VI.F. Exercises

F.I. Let g be a loxodromic element of the discrete subgroup G of P", and let xbe a fixed point of g. Then x is a point of approximation.

F.2. (a) There is a Kleinian group G, , and a point x e A(G, ), where x is both apoint of approximation and a fixed point of an elliptic element of Gt.

VI.F.Exercises 133

(b) There is a Kleinian group G2, and a point xeA(G2), where x is a fixedpoint of an elliptic element of G2, and x is not a point of approximation.

F.3. Let G be a geometrically finite discrete subgroup of M, and let H be asubgroup of finite index in G. Then H is geometrically finite.

F.4. Let G be a finitely generated non-elementary Fuchsian group of the secondkind, and let A be a boundary axis of G. Then Stab(A) is hyperbolic cyclic.

FS. Let G be an elementary Kleinian group. Then G is geometrically finite.

F.6. Let G be a geometrically finite Kleinian group where oce°Q(G). Then forevery limit point x, which is not a parabolic fixed point, there is a g e G withIg'(x)I > I.

F.7. Let G be a Kleinian group where oc E Q(G) and G contains no parabolicelements. Suppose that for every x e A, there is a g e G with Ig'(x)I > 1. Then Gis geometrically finite (also see VIII.H.5).

F.8. Let G be a Kleinian group with oc e'12. Suppose that x is a rank I cuspedparabolic fixed point. Then there is a point y, equivalent to x, and there is acusp E at y, where E lies outside all isometric circles of G.

F.9. Let G be a geometrically finite Kleinian group. Then the 2-dimensionalLebesgue measure, meas(A(G)) = 0. (Hint: use B.9.)

F.10. Let G be a non-elementary geometrically finite Fuchsian group acting ona-B2, where j(z) = z + I E G. Let x be a point of approximation for G. Then thereis a sequence {g,,,} of distinct elements of G, with g.(oo) # oo, and there is aconstant K > 0, so that Ix - where p,,, is the radius of theisometric circle of g,,,.

F.11. Apply F.10 to the modular group, PSL(2; Z), and show that if x is irrational,then there is a sequence {d/c} of distinct rational numbers, and there is a constantK > 0, so that Ix - d/cl 5 K Icl-2.

F.12. Let G be a Kleinian group. For every maximal Euclidean subgroup J of G,set a(J) = 0, if J has rank 2; a(J) = 1, if J is cusped but not doubly cusped;a(J) = 1, if J is doubly cusped and not cyclic; a(J) = 2, if J is doubly cusped andcyclic. Assume that there are only finitely many conjugacy classes of rank 1maximal Euclidean subgroups of G, and let fi = Y a(J), where the sum is takenover a maximal set of non-conjugate such subgroups. Then there is a conformalembedding of Q(G)/G into a Riemann surface S, so that (S - Q(G))/G consistsof exactly # distinct points.


VI.G. Notes

A.5. See the note to J.17 in IV.K., also see Apanosov [9] or Wielenberg [97].A.15. Essentially finite fundamental polyhedra (more precisely, the correspon-ding manifolds) were first discussed by Marden [52]. The impressive machinerybuilt up by Thurston to understand these manifolds and orbifolds is beyondthe scope of this book; a description of some of this machinery can be foundin Thurston [91] or Morgan [74]. B.1. The term "point of approximation" firstappeared in [12]. These points were first studied by Hedlund [32], and aresometimes known as Hedlund points. C.7. The equivalence of parts (i) and (iii)is due to Reardon and Maskit [12]; the equivalence of(i) and (ii), for torsion-freegroups, is due to Marden [52]. D.1. Specific examples exhibiting global structuresof hyperbolic manifolds and orbifolds can be found in Vinberg [96], Riley [81],[82], [83], Thurston [90], and others. E.7. This was first observed by Yamamoto[ 102]. F3. See [74 pg. 71 ] for a proof of Thurston's theorem that every finitelygenerated subgroup of a geometrically finite Kleinian group is geometricallyfinite. F.9. This fact is due to Ahlfors [6]; Bonahon [18] has recently proved thatthe measure of the limit set is zero for every finitely generated Kleinian groupthat cannot be decomposed as a non-trivial free product.

Chapter VII. Combination Theorems

There are two distinct but related combination theorems: the amalgamated freeproduct, and the HNN extension. The basic outlines of these, in a purely abstractsetting, are given in sections A and D. For Kleinian groups, the purely abstractsetting is sufficient to prove that the combined group G is discrete and hasthe named group theoretic structure, but does not suffice to give a clear under-standing of Q/G or of 1113/G; nor does it yield sufficient information to decidewhether or not G is geometrically finite. The necessary inequality is given insection B. The combination theorems themselves (these are sometimes known asthe Klein-Maskit combination theorems) are given in sections C and E. We stateand prove these theorems only for discrete subgroups of M. The minor modifica-tions required for the case that G contains orientation reversing elements are leftto the reader.

We state and prove the combination theorems in sufficient generality toinclude their major uses in dimensions 2 and 3; these include Thurston's uni-formization theorem, and the classification of finitely generated function groups(see Chapter X).

Specific constructions of Kleinian groups using combination theorems appearin Chapters VIII and IX. We also use combination theorems for more theoreticalpurposes in Chapters IX and X.

Throughout this chapter, all groups are subgroups of some universal groupG, which will always be the group of homeomorphisms of some space X ontoitself; we will often specify the space X as C, and the group G as M. Also,throughout this chapter, the index "m" has the range of values: [1, 2}.

VILA. Combinatorial Group Theory - I

A.I. For any group G, if G, , G2, ... are subgroups of G, and g, , 92,... are elementsof G, then <G G2, ... , g, , g2, ...) denotes the smallest subgroup of G containingall the subgroups G,, G2, ... , and containing all the elements g, , 92, ... .

A.2. In this section, our basic hypothesis is that we are given two groups, G, andG2, with a common subgroup J. We also assume throughout that [G.,: J] > 1.

136 VII. Combination Thcorcros

A normal form is a word of the form g.... g where either every gk with evenk lies in G, - J, and every gk with odd k lies in G2 - J, or vice versa; that is, fork even, gk a G2 - J, and fork odd, g,k e G, - J.

There is a natural identification of normal forms as follows. If j e J, then weregard the forms g... - gk... g, and g,, ... (g k j) (J - t gk-t) . . . g, as being equivalent.

There is an obvious multiplication of normal forms, formed by juxtapositionof words, and then contraction, using the above equivalence. The product of twonormal forms is equivalent to either a normal form, or to an element of J. Theset of equivalence classes of normal forms, with this multiplication, together withthe elements of J, is called the free product of G, and G2, with amalgamatedsubgroup J, or sometimes, just the amalgamated free product, and is written asG, *, G2.

A.3. There is a natural homomorphism 0: G, *j G2 -+ G = <G,, G2> given by re-gardingjuxtaposition of words as composition of mappings; that is, 0(g.... g,) =g. o . o g, It is clear that equivalent normal forms are mapped onto the sametransformation.

If 0 is an isomorphism, then we say that G = G, *, G2, and we do notdistinguish between G and G, *,, G2. In this case, we regard normal forms as beingelements of G; and we write them as composition (a o b), rather than juxtaposition(ab).

Since J is embedded in both G, *.r G2 and G, it is easy to see that G = G, *, G2if and only if 0 maps no non-trivial normal form to the identity.

A.4. Every normal form has a length, n = Ig - g, 1. Note that equivalent normalforms have the same length, so if G = G, *, G2, then IgI is well defined for everyelement of G.

If G = G, *, G2, and J is trivial (in this case we say that G is the free productof G, and G2, and we write G = G, * G2), then every non-trivial element of Ghas a unique normal form, while if J is non-trivial, the normal form of an elementof G is clearly not unique.

A normal form g.... g, , is called an m -form, if g a G. - J. Every normal formis either a 1-form or a 2-form. This concept is also invariant under equivalence.

AS. Using normal forms, it is easy to see that isomorphisms can be extended.That is, suppose we have two amalgamated free products G = G, *, G2, andC = C, *1 62, and suppose there are isomorphisms cps,: G. - 0., where (p, IJ =cp2I J = 7. Then there is a unique isomorphism cp: G -+ C, where cp I G. = cps,.

A.6. If G = <G1, G2 > acts freely and discontinuously on X, then there is a criterionfor G to be the amalgamated free product.

An interactive pair of sets (X X2), consists of two non-empty disjoint setsX, and X2, where X. is invariant under J, every element of G, - J maps X, intoX2, and every element of G2 - J maps X2 into X1.

VILA. Combinatorial Group Theory - 1 137

Note that if (X,, X2) is an interactive pair, then X. is precisely invariant underJ in Gm. While there can be pairs of sets (X,, X2) where X. is precisely invariantunder J in Gm, but the pair is not interactive; there is one important special casewhere the pair is necessarily interactive.

Proposition. Let X = §", G = Pn+1, and let W be a topological (n - 1)-spherebounding two open topological discs, X, and X2. If X. is precisely invariant underJ in Gm, then (X,, X2) is an interactive pair.

Proof. Let g be some element of G, - J. Since X. is J-invariant, so is Xm. Forevery g e G, - J, g(X,) fl x, = 0; hence g(W) fl x, = o. It then follows that9(X I) c X2. Similarly, if g e G2 - J, then g(X2) c X1 .

A.7. Theorem. If G acts freely and discontinuously on (some non-empty open subsetof) X, and G = G, *,, G2, then there exists an interactive pair of sets.

Proof. Let D be a fundamental set for the action of G on °Q(G) X. SetX, = U g(D), where the union is taken over all 2-forms, and set X2 = U g(D),where the union is taken over all 1-forms. It is immediate that both X, and X2are non-empty and J-invariant.

Suppose there is a point x c- X, fl X2. Then there are points y, and y2 in D,and there is a 1-form h,, so that x = h,(y,), and there is a 2-form h2, withx = h2(y2). Since y, and y2 are G-equivalent points of D, y, = Y2 = Y. Writeh, = g.- -g,, and write h2 = fk o ... of, . Then since h' o h2(y) = y,

1 =h'oh2 =91o...ogn'of 0...of,.

Since g" a G, - J, and fk E G2 - J, the expression above is a normal form; henceit is not the identity in G. We have shown that X, fl X2 = 0.

If x is a point of X2 then there is a y e D, and there is a 1-form h 1 = g" o- o g l,so that x = h1(y). If h2 is any element of G2 - J, h2(x) = h2 o g" o o g1(y); since

h2 o g" o o g, is a normal 2-form, h2(x) a X1 . Similarly, if x e X1, and h, e G, - J,then h,(x)e X2.

A.8. The converse to the above theorem is not quite true. Let j(z) = z + 1, andlet J = <j>. Let g, (z) = -z, and let g 2, a(z) = -z + 2a, a >0. Let G, = < j, g, ).and let G2,a = 0192,.X

Note that G, is the (2, 2, cc)-triangle group; that is, it is the non-cyclic rankI Euclidean group, and that G2,a is a conjugate of G, in M. For every a, the upperhalf plane, X,, is precisely invariant under J in G,, and the lower half plane, X2,is precisely invariant under J in G2,a. Hence, (X,, X2) do form an interactive pairof sets for G, and G2,a. However, if a is irrational, then G. = <G1, G2,. > is notdiscrete, while if a is rational, then there is an integer a, so that (g, o 92,.)'C- J; i.e.,G. is not the amalgamated free product.

138 VII. Combination Theorems

A.9. We return to our basic assumption that J is a subgroup of both G1 andG2; and we set G = <G1,G2). For our next application, we need the notion ofan (m, k)-form. The normal form g.... g, is an (m, k)-form if g e G. - J, andg1eGk-J.

An interactive pair (X1,X2) is proper if either there is a point in X1 that isnot G2-equivalent to any point of X2, or there is a point in X2 that is notG1-equivalent to any point of X1.

Lemma. If there is an interactive pair (X1, X2), and g = g.... g, is an (m, k)-form,then P(g)(Xk) C X3-m. Further, this inclusion is proper if (X1,X2) is proper, andIgI>2.

Proof. If n = 1, then k = m, and gl(Xk) c X3 _k. Let g.... g, be a normal (m, k}form; assume that fi(g,,...g1)(Xk) = 9.o."o91(Xk) c X3-m, and thatG3-m - J. Then 9.+1(X3-m) c Xm, so 9n+1 og o...o91(Xk) c 9N+1(X3-m) a X.-

Assume for simplicity that the G1-translates of X1 do not cover X2. Ifg,eG1 - J, then g1(X1) is properly contained in X2. Theng o o 92(X2) c X3_m Since the first inclusion is proper, so is the combinedinclusion. If g1 a G2 - J, we might have that 9i(X2) = X1, but then 92 a G1 - J,so 92(X,) is properly contained in X2. Then 92 o 91(X2) is properly contained inX2. Hence, as with the previous case, g o o g2 o gi (X2) is properly containedin X3-m. 0A.10. Theorem. Assume that there is a proper interactive pair of sets (X1, X2) forthe groups G, and G2 with common subgroup J. Then G = <G1, G2 > = G1 s, G2.

Proof. It is immediate that no normal form of length one can be the identity inG. If g = g,,... g

1is a normal form, with n > 1, then by A.9, ds(g)(X) is properly

contained in either X1 or X2, where X is either X1 or X2. In any case, 0(g):0 1.0

A.11. Given the precisely invariant set Xm, a fundamental set D. for G. is maximalwith respect to X. if D. fl Xm is a fundamental set for the action of J on X.-

A.12. Theorem. Let the discontinuous groups G1 and G2 have a common subgroupJ. Assume that there is an interactive pair of sets (X1,X2), and assume that thereis a maximal fundamental set D. for Gm, so that for every g e Gm, g(D, nX3_m) CX 3_m. Let D = (D1 fl X2) U (D2 fl X, ). Then D is precisely invariant under theidentity in G.

Proof. There is nothing to prove if D = 0. Assume there is a point x e D1 fl X2;the case that x e D2 fl X1 is handled analogously.

If g is a non-trivial element of J, then since X2 is J-invariant, g(x) a X2. Sinceg is a non-trivial element of G1, g(x) 0 D,. Hence g(x) # D.

VII.B. Blocks and Spanning Discs 139

If gEG, - J, then by the maximality of D,, g(x)EX2; of course, since g isnon-trivial, g(x) 0 D,. We have shown that g(x) E X2 - D1.

If g e G. - J, then since g(X2) c X, , g(x) E X, . Since D2 is maximal, everypoint of X2 fl °Q(G2) is G2-equivalent to some point of D2 fl X2; since x is in X2and g(x) is in X,, g(x) cannot be in D2. We have shown that g(x) a X, - D2.

Now let g = g.... g, be a normal form. We have already taken care of thecase that I gI = 1; assume that I g I > 1, and assume that if is an m-form withjhi < n, then 0(h)(x)EX3_m - D,.; this is consistent with the statements abovefor n = 1.

Let h = g _, o o g, . Assume first that his a 1-form; i.e., that g._, e G, - J.Then h(x)EX2 - D,. Since g is a normal form, J, so g(X2) c X,; inparticular, g(x) a X, . Since D2 is maximal, no point of X2 is G2-equivalent to anypoint of D2 fl X, . Hence g(x) 0 D2. The proof in the case that E G2 - J isessentially the same. El

A.13. One can view Klein's combination theorem as an application of A.10 andA.12. The original theorem was stated in terms of two Kleinian groups whereeach has a fundamental domain containing the exterior of the other.

Theorem. Let G, and G2 each act freely and discontinuously on some open subsetof X. Suppose there is a fundamental set D. for G., where D, U D2 = X, andD=D, fl D2 0 0. Then G = (G,, G2> = G, * G2, and D is precisely invariantunder the identity in G.

Proof. If either G, or G2 is trivial, there is nothing to prove; hence we can assumethat they are both non-trivial. Then D, - D2 and D2 - D, are both non-empty.Set X,=D,-D2,and set X2 = D2. Since D,UD2=X,X,UX2=X.If gis anon-trivial element of G,, then g(D,) fl D, = 0, so g(X,) c g(D,) c D2 = X2.Also if g is a non-trivial element of G2, then g(D2) fl D2 = 0, so g(X2) = g(D2) cX - D2 = D, - D2 = X,. We have shown that (X X2) is an interactive pair ofsets for the groups G, and G2, with the identity as the common subgroup.

If x is any point of D, then x # X, . Since x c- D, , x does not lie in anyG,-translate of X,. Hence the interactive pair is proper. By A.10,

Note that since D,, fl X,, =X., D. is maximal. Also, D, fl X2 = D, fl D2, andfor every g(D, fl D2.) c D2 = X2. The last hypothesis of A.12 is triviallysatisfied, for D2 fl X, = 0. It follows from A. 12 that D is precisely invariant underthe identity in G. 0

VII.B. Blocks and Spanning Discs

M. Throughout this section, J is a geometrically finite subgroup of the non-elementary discrete group G in M, and B e (C is closed and J-invariant. The main


example that one should keep in mind is that J is a finitely generated Fuchsiangroup, not necessarily of the first kind, and that B is either a closed invarianttopological disc, or the boundary of such a disc. Another important example isthat J is any geometrically finite subgroup of G, and B = 4(J). In any case,BDA(J).

Recall that a cusp C at the fixed point x of the primitive parabolic elementj e G, is any set of the following form. Let g e M be such that g o j o g-' (z) = z + 1;then C = g-'({zJO S Re(z) < I,lm(z) > 1}).

B.2. Proposition. For a closed J-invariant set B, the following statements areequivalent.

(i) B fl Q(G) = B fl Q(J), and B fl Q(J) is precisely invariant under J in G.(ii) there is a constrained fundamental set E for J, containing a fundamental set

D for G, there are a finite set of disjoint cusps C1,..., C. at parabolic fixed pointsof J, where each Cj c E, and q fl B = 0, and there is a neighborhood U of B sothat (E - D) fl u c C, u . U C,,.

Proof. We first assume (i). Let E be the fundamental set for J given as theboundary of a Dirichlet fundamental polyhedron; then E is constrained, E hasfinitely many sides, and, except perhaps for some parabolic cusps, E is boundedaway from A (J). If B = A (J), then condition (ii) is automatically satisfied, for Eitself is bounded away from B, except perhaps for some cusps. We now assumethat B fl Q(J) = B fl Q(G) s 0; in particular, we assume that G is Kleinian.

Since B fl o(G) = B fl Q(J) is precisely invariant under J, a subset of B fl Q(J)can be mapped into itself only by an element of J; in particular, E fl B is containedin °Q(G) and is precisely invariant under the identity in G. We set D fl B = E fl B.

Suppose (ii) is false; then there is a sequence of points {zk } in °Q(G) fl E,with zk - z e B, and there is a sequence of non-trivial elements {gk} of G, with9k(zk) a E, and gk(zk) -+ z' e B.

We permit E - D to contain a finite number of cusps at parabolic fixed points.Aside from these cusps, E is bounded away from 4(J); hence we can assume thatz e Q (J); likewise, we can assume that z' e Q(J). Let N be a nice neighborhoodof z; we can assume that the zk all lie in N.

If the elements { gk } are all distinct, then dia(gk(N)) -+ 0, so z' is a limit pointof G in B. Since B fl Q(G) = B fl Q(J), z' is a limit point of J. But z' e D(J).

If the elements { gk } are all equal, gk = g, then g(z) = z'. Since z and z' are inB, and B fl Q(J) is precisely invariant under J, g e J. Since z and z' are both in E,z = z' and g = 1, contradicting our assumption that the gk are all non-trivial.

B3. We next assume (ii) and prove (i). We are given that E fl B = D fl B; since theJ-translates of E fl B cover °Q(J) fl B, Q(J) fl B = °Q(G) fl B. Let z e Q(J) fl B.Since E is constrained, there is a neighborhood N of z, and there are a finitenumber of elements j 1, ..., j of J so that U jk(E) covers N - {z}. Since z e B, ifwe choose N sufficiently small, the sets jR' (N) fl E all lie in both U and D. Hence,

VII.B. Blocks and Spanning Discs 141

there is a neighborhood N of z so that N is precisely invariant under Stabj(z) inG. We have shown that B fl S2(J) c B fl SA(G); the opposite inclusion is trivial.

Suppose there is some g e G, and there are points x and y in B fl Q(G) withg(x) = y. We saw above that there is a neighborhood N of x so that every pointof N - {x} is J-equivalent to some point of D n u c E; likewise, there is aneighborhood M of y so that every point of M - {y' is J-equivalent to somepoint of D n U. We can assume that g(N) = M, from which it follows that g e J.

D

B.4. A closed J-invariant set B is a block, or (J, G)-block if we need to specify Jand G, if it satisfies either (and hence both) of the conditions above, and alsosatisfies the following: for every puncture on Q(J)/J, there is a neighborhood Uof the puncture (that is, U is a punctured disc), so that either U is contained inthe projection of B, or U is disjoint from the projection of B.

One can restate the above condition as follows. If C is a cusped region forthe parabolic element j E J, then there is a cusped region C c C with eitherC'cB,orC'fB=0.

Since J is geometrically finite, if J has no parabolic elements, then there is aconstrained fundamental set E for J which is bounded away from A(J); in thiscase, one can replace (ii) by the condition:

(ii') (For the case that J contains no parabolics.) There are constrainedfundamental sets D for G and E D for J, and there is a neighborhood U of B,with D fl u = E fl U.

However, if J has parabolic elements, there might be a parabolic fixed pointx in B where Stabc,(x) has rank 1, and x is doubly cusped in J, but not in G.Then any constrained fundamental set E for J will contain a cusp C, centered atsome point y J-equivalent to x, where C contains limit points of G accumulatingto y (see VIII.H.3).

For some of our applications, we need a somewhat stronger concept. A blockis strong if for every rank I parabolic fixed point x of J, either StabG(x) has rank2, or x is doubly cusped in G.

B.5. There are two special cases of blocks which should be mentioned. First, Bmight contain a parabolic fixed point x of J, where x is also a fixed point of aparabolic or elliptic element g e G - J, and x e g(B) n B (for examples of this, seeVII.A.8, or VIII.G.1-3). Similarly, J might contain a loxodromic cyclic subgroupJ0, with fixed points x and y, and there might be a loxodromic or elliptic elementg e G - J, where g also has fixed points at x and y. Then, except for these twopoints, g(B) is disjoint from B (see VIII.E.4, 8).

B.6. Assume now that B is a block, and that E is a constrained fundamental setfor J as in B.2(ii). That is, there is a fundamental set D c E for G, and there area finite set of cusps, C1, ... , C. in E, so that, except for these cusps, E is bounded

142 V11. Combination Theorems

away from A (J), and each Ck either is contained in D, or is G-equivalent (but notJ-equivalent) to a cusp in D, or contains limit points of G.

The cusps of E near B, that are not contained in D, are called excludedcusps; since B either contains, or is disjoint from any sufficiently small cuspedregion, the excluded cusps can be chosen so that they do not intersect B. Anexcluded cusp C, with vertex x, is a rank 2 exceptional cusp if StabG(x) has rank2; of course Stab,(x) has rank 1. An excluded cusp C, with vertex x, is a rank 1exceptional cusp, if there is a g e G - J, and there is a non-excluded cusp C c D,so that every point of C near x is J-equivalent to some point of g(C'). It is easyto see that C is a rank 1 exceptional cusp if and only if there is a g e G - J, whereg(x) E B is also a parabolic fixed point, and g(C) is J-equivalent to a non-excludedcusp in E.

One now easily proves the following.

Proposition. A block is strong if and only if every excluded cusp is exceptional.

B.7. It is worth noting that if there is a constrained fundamental set E for J,containing a fundamental set D for G, and E - D is bounded away from B, thenB is strong. A particular case of this occurs for example when J is a Fuchsiangroup of the first kind, acting on 0-02, where Q-12/J is compact, and B is either theclosure of H2, or the circle at infinity.

It is also immediate from the definition that if B is a (J, G)-block, and G isgeometrically finite, then B is automatically strong.

B.8. In general, if B c C is J-invariant, and C c H' is also J-invariant, withdC = B, then we say that C spans B provided it satisfies the following condition.For every rank I parabolic fixed point in B, there is a doubly cusped region A U A'so that C does not intersect either of the two half-spaces whose boundaries are,respectively, A and A'. In the special case that B is a simple closed curve and theopen disc C c 4'4' is precisely invariant under Stab(B), we call C a spanning disc.

In the theorem below, we want to include the possibility of a spanning setfor B equal to B itself. A (Euclidean) closed set C c is' weakly spans the blockB c S2 if C is J-invariant, C fl S2 c B and for every rank I parabolic fixed pointof J in B, there is a doubly cusped region A, so that C does not intersect eitherof the two disjoint half-spaces which together span A; in particular, B weaklyspans B.

B.9. Theorem. Let G be a discrete subgroup of P3, let B c §2 be a (J, G)-block,and let C e 4'33 weakly span B. Let G = EgkJ be a coset decomposition. Then

dia6(gk(C)) < co.

Proof. Normalize G so that the origin in 033 is not fixed by any element of G.Since reflection in S2 commutes with every element of G, and interchanges 0 andoc, Stab(0) = Stab(co) = { I }. Let E be a fundamental set for the action of J on1'' obtained from the Ford region by adjoining some points on the boundary.

VILB. Blocks and Spanning Discs 143

Since t fl 033 is the Dirichlet fundamental polyhedron, t (1133 is an essentiallyfinite fundamental polyhedron.

Let g be some element of G - J; observe that (goj)-'(co) = j-' og-'(oo), soif we fix g, and let j vary through J, the set of points {(g o j)-' (oo)} is J-invariant.For each k, choose the coset representative gk, so that ak = gkI(oo), the center ofthe isometric sphere of gk, lies in E.

Let 0. denote the exterior of 033, together with the point at oc, so that I_ is alsoa model of hyperbolic 3-space, and let 9 = E fl L.

Since L is essentially finite, outside of a finite set of horoballs and extendedhoroballs, Ti , ..., T,;, EP is bounded away from A(J). Choose these horoballssufficiently small so that none of them contain any G-translate of Co. An extendedhoroball in I. is the union of a horoball and two disjoint half-spaces, where eachhalf-space spans a cusped region. Choose these extended horoballs sufficientlysmall so that none of their half-spaces intersects C. For each extended horoballT', let T. be a smaller extended horoball contained inside T;, where the half-spaces of T span cusped regions that are properly contained in the cusped regionsfor T'.

We sum separately over those k for which ak lies outside all these extendedhoroballs, T ..., T,,, (the horoballs themselves contain no ak, for each ak is aG-translate of oo), and over those k for which ak lies in one of the half-spaces Hcontained in an extended horoball T = Tk. Let H' H be the correspondinghalf-space of Tk.

For those k for which ak lies outside all the extended horoballs, since E iscontained in I_, and, outside these horoballs, is bounded away from A(J), theseak are uniformly bounded away from C. Hence for k sufficiently large, theisometric sphere for gk is disjoint from C, so gk(C) is contained inside the isometricsphere for gk'. The result now follows from IV.G.7.

We now assume that the ak all lie inside the half-space H, with vertex x. SinceEuclidean rotation does not distort distances, we can assume that x is at theNorth Pole of V. Let S be the sphere of radius one about x, and let r denotereflection in S.

Since we need an extended horoball for E near x, Stabj(x) has rank 1; let jbe a parabolic generator of Stab,(x).

We now write points in E' as (z, t), z e C, t e R, and think oft as height. Thatis, we consider the coordinates of a point ye E' as being Im(y), Re(y), and height,t. Since S is centered at the North Pole (i.e., x = (0, 1)), r(IB3) is the half-spaceof height less than 1/2. Since r o j o r preserves r(B3), it is a translation of the form(z, t) - (z + fi, t); if necessary, we conjugate by another Euclidean rotation so thattg is real, and so that r(H) is the set {ye ElIIm(y) > b}. Then r(H') is of the form{ ylIm(y) > b'), where b' < b.

Since C intersects the span of H' only at x, r(C) is contained in { y I lm(y) < b' }.The points {ak } are all contained in a cusp centered at x, and they lie outside

a horoball centered at x. This means that the points {r(ak)} have height boundedbetween 1/2 and some upper bound; their real parts are bounded above andbelow, and their imaginary parts are bounded below by b.


Let pk be the radius of the isometric sphere of gk.

B.10. Lemma. There is a constant K1 > 0 so that pk/lak - xI <- K1.

Proof. Since x is a parabolic fixed point, there is a geG, so that g(x) lies on 8D,where D is the Ford region for G. Every h e G is conformal, so the local distortionof distance at a point y e f' is independent of direction, and can be computed asIh'(y)I, the norm of the differential of h at y. Since g(x) lies on or outside everyisometric sphere, lh'(g(x))l S 1 for all heG. Since Ih'(g(x))I = I(hog)'(x)I/Ig'(x)I,we conclude that there is a constant K' so that Ih'(x)I S K', for all heG.

Since gk is the composition of reflection in its isometric sphere, followed bya Euclidean motion, it is clear that Ig'(x)I depends only on pk and on Iak - xl.Since these maps are all conformal, the relationship is independent of dimension.We compute in dimesion 2. Write gk(z) = (akz + bk)/(ckz + dk), akdk - bkck = 1,and obtain Ig;<(x)I = Ickx + dkl-2 = Ickl-Zlx + dk/ckl-2 = pk2/lak - x12.

We have shown that pk2/lak - x12 = Igk(x)1 < K'. Now set K1 = K'.

B.11. Lemma. There is a constant K 2 > 0 so that for every ak a H, and for everyyeC,Ia. - yl -K2lak-xI.

Proof. We first need to observe that if a and y are any two points, distinct fromx, and r denotes reflection in the sphere of radius one about x, then I r(y) - r(a)I =ly - al/ly - xl Ia - xI. To prove this, let P be the plane determined by x, y anda; then r preserves P. We can think of P as being the complex plane and computer(z) = 1 /(a - x) + x, from which the desired equality easily follows.

Apply this to a = a,, to obtain

Iak - yl/lak - xl = I r(ak) - r(y)l Iy - xl= Ir(ak) - r(y)I/I r(y) - xl.

Since the imaginary part of r(ak) - r(y) is bounded from below by b - b', weonly need to consider the case that r(y) - oo. Since r(ak) has bounded height, theheight of r(y) - r(ak) is commensurable with the height of r(y) - x. Similarly, thereal part of r(y) - r(ak) is commensurable with the real part of r(y) - x. SinceIm(r(ak)) > b, and Im(r(y)) -+ - oo, there is a positive constant K so that

IIm(r(y)) - Im(r(ak))I >- Kllm(r(y)) - lm(x)l.

The result now follows.

B.12. We now conclude the proof of B.9. Let Sk be the distance from ak to C; recallthat pk is the radius of the isometric sphere ofgk. Then Sk is the infimum of Iak - ylforyeC,sobyB.ll,Sk?K21ak-xl.

Using the technique of IV.G.7, we obtain

dia(gk(C)) < pk/Sk < K21pk/Iak - xj.

VILB. Blocks and Spanning Discs 145

By B.1O, the right hand side of the above is bounded by (K,/K2)pk. Finally, weknow from IV.G.7 that >(pk)6 < oo

B.13. Corollary. Let J be a geometrically finite subgroup of the discrete groupG e P'. Let G = >g,,J be a coset decomposition. Then >dia6(gk(A(J)) < oo.

8.14. Corollary. Let B be a (J, G)-block, let C weakly span B, and let {g,k(C)} bean enumeration of the distinct translates of C, then

dia(g,k(C)) - 0.

B.15. Let B be both a strong (J, G)-block, and a simple closed curve. Then Bdivides aR into two open topological discs. We say that B is precisely embeddedif for every g e G - J, g(B) does not intersect one of these two open discs. If B isprecisely embedded, then while some g(B) may have points in common with B,it does not cross B.

B.16. Theorem. Let G be a discrete subgroup of M; let J be a geometrically finitesubgroup of G; let W be a simple closed curve in C, where W is a precisely embeddedstrong (J, G)-block. Then there is a spanning disc C e H3 for W.

B.17. Proof. First of all, there is some disc C c H' spanning W. For example,choose C to be one of the boundaries of the convex hull of W Since C is a disc,and J is geometrically finite, hence finitely generated, C/J is a surface of finitetopological type; however, this surface need not be at all smooth, and need notbe embedded in H3/G (it is clear that C is precisely invariant under J if and onlyif C/J is embedded in 193/G).

Let M = (H3 U Q(G))/G, so that M is a three manifold, possibly with bound-ary; if G is Kleinian, then 8M # 0. Let S be the image of C/J in M. We needto show that S can be deformed inside M so that it becomes an embedded surface.A point of self-intersection of S is a singular point; a point of C lying over asingular point, is also called singular.

We first observe that the singular points of C are bounded away fromw fl ow = W fl Q(G). Let z be some point of w fl Q; assume that there is asequence of elements (9k) of G, and there is a sequence { xk } of points in gk(C) fl c,with xk -+ z. For any given g e G, g(W) does not cross W, and g(W) fl W is disjointfrom Q(J). Therefore, there is a subsequence so that the {gk(W)} are all distinct.By B.14, dia(gk(C)) - 0; contradicting the fact that z e Q(G). We restate this asfollows.

Lemma. There is a neighborhood U of 8S in M, so that S fl U is embedded in M.

B.18. Since J is geometrically finite, there is a fundamental polyhedron E for Jso that E is bounded away from A(J), except for a finite set of possibly extendedprecisely invariant horoballs. Since J preserves a simple closed curve, everyEuclidean subgroup has rank 1; hence the horoballs are all extended. We sawabove that the singular points of C are bounded away from f?(J) fl W; hence


there is a finite set of extended horoballs whose union, together with a compactsubset of 0-03, contains all the singular points of E.

B.19. Using standard techniques in 3-dimensional topology, we can assume thatS is smooth, that there are a finite set of smooth curves on S containing all thesingular points, and that each of these singular curves is just a simple crossing(that is, locally, if V is a singular curve on C, then there is exactly one translateg(C) which intersects C along V). Of course, these singular curves on S need beneither simple nor disjoint; this is the source of our difficulties. Pick one of thesecurves, call it v; then there is a g e G, so that g(C) and C intersect in a curve Vlying over v (the projection of g(C) fl c may contain singular curves other thanv). The stabilizer of V can be either loxodromic, elliptic, or parabolic cyclic, ortrivial.

B.20. We first deform C inside the extended horoballs, and eliminate the possi-bility that some singular curve has parabolic stabilizer. Let x be the fixed pointof the primitive parabolic element jeJ, and let T be an extended horoball forStab(x) in J. Normalize so that x = oo, and so that j(z) = z + 1; then we canassume that t3T is at height one. It is clear that we can deform C to a new surfaceC, so that c, n 8T is a single j-invariant curve Wt. Let h(a, r) be a j-invariantdeformation of W, to the straight line W. = {(z,t)e H3IIm(z) = 0, t = 1}; that is,h(u, 0) = W, (a), h(a, 1) = W2(a), and h(a + 1, r) = h(o, r) for all T. Inside T, re-place C, as follows. For 1 < t < 2, define C2(s, t) = (h(s, t - 1), t); for t >- 2, let C2be the Euclidean half-plane: {Im(z) = 0, t >- 2}. Now define C2 outside T, so thatit is J-invariant and so that it agrees with C, outside all the translates of T

After performing the above operation on the finite number of non-conjugateextended horoballs, we have a new surface C, which still spans W, and now allthe singular points are contained in a compact subset of S.

B.21. We say that W weakly separates g,(W) and g2(W) if g,(W) lies in one ofthe closed discs bounded by W, and g2(W) lies in the other. For every g e G - J,we define the distance from g to J to be one plus the number of distinct translatesof W that weakly separate W from g(W); we write this distance as IgJ. Since thediameter of any sequence of distinct translates of W tends to zero, the distanceis always finite. It is easy to see that for every jeJ, the distance from g to J is thesame as the distance from jog to J.

Modulo J, there are only finitely many singular curves on C. Let V,,..., V.be a complete list of inequivalent such singular curves. For each Vk, there is agk e G so that gk(C) and C intersect along Vk; note that the translate gk(C) dependsonly on Vk; likewise, Igkl depends only on Vk. Also, if we replace Vk by anothersingular curve with the same projection, then we replace gk by an element of theformjogk,and Igkl = Ijogkl.

There is a k so that Igkl is maximal; set g = gk. Since neither C nor C _g(C) has any self-intersections, the intersection of C with C in 093 consists of

V11.B. Blocks and Spanning Discs 147

a set of simple loops and arcs, where each loop is stabilized either by the iden-tity or by a common maximal elliptic subgroup of J and gJg-'. Similarly,each arc is stabilized by a common loxodromic cyclic subgroup of J and gJg-'.Two of these arcs stabilized by the same loxodromic cyclic group have theirendpoints on W in common; otherwise, the arcs and loops comprising C fl C areall disjoint.

This set of singular curves is of course invariant under J = J fl gJg-', andmodulo J, their number is finite. For any loxodromic cyclic subgroup of J, thenumber of arcs stabilized by that subgroup is finite. Since W and g(W) = donot cross, the number of singular arcs of intersection of C and C stabilized byany loxodromic cyclic subgroup of J is even.

Each singular loop bounds a disc on C; likewise, each pair of arcs stabilizedby the same loxodromic cyclic group bounds a disc on C, where now theboundary of the disc includes the fixed points of the stabilizer of these arcs.Choose a singular loop V, or pair of singular arcs V and V', so that the discM c C bounded by this loop, or pair of arcs, contains no other points ofintersection with C.

The disc C divides D-I into two topological half-spaces. One of them, calledthe outside of C contains 14' in its boundary. The other half-space is the inside ofC. Likewise, the outside of C has W on its boundary. and the inside of C doesnot have all of W on its boundary.

We now have two cases to consider, according as M lies inside or outside C.

B.22. We first take up the case that M lies inside C. If M is bounded by a singleloop V, then M intersects only finitely many translates of C. Hence we can choosea parallel surface M, where Si is so close to M that it is still an embedded disc,and Si crosses exactly those translates of C that Si crosses. There are two possiblechoices of direction for Si, we choose it to lie outside C. We use hyperbolicdistance to define the parallel surface, so that Si and a are both invariant underStab(M). It is not clear where Siais, we cut it off or extend it as necessary, sothat l = Sialies on C.

If Si is bounded by a pair of arcs V and V', then Stab(V) = Stab(V') isloxodromic cyclic. In this case, we start with a fundamental polyhedron E forStab(V) = Stab(M). Observe that there is a neighborhood of Si in £ that inter-sects only finitely many translates of C. Hence we can find a (hyperbolically)parallel surface Si, again lying outside C, so that Al is precisely invariant underStab(M), Al is an embedded disc, and Al intersects exactly those translates of Cthat Si intersects. As above, we extend or cut off the boundary of Al so thatOS? = 17 lies on C.

Let M be the disc on C bounded by P, where M contains the disc boundedby the loop V, or pair of loops V and V'. We form a new disc C as follows. ReplaceM by Si, and for every j e J, replace j(M) by j(A1). Since Si does not intersect C,neither does Si, so C is again an embedded disc. We have defined C so that it isinvariant under J. Also, if h(W) lies inside C, then the distance from h(C) to g(C)


(i.e., Ih' ogl) is greater than Igi, so h(C) and g(C) do not intersect; in particular,there is no intersection of P. with h(C). If h(W) lies outside C, and h(C) intersectsft, then since M lies inside C, h(C) also intersects C. We have shown that if h(C)intersects C, then h(C) intersects C. Of course, for some h, we may have increasedthe number of curves of intersection modulo J, but the number of curves ofintersection of g(C) with C, modulo J, is less than the corresponding number ofcurves of intersection of g(C) with C.

B.23. We next take up the case that M lies outside C. Let M be the disc on Cbounded by the same loop or pair of arcs, and consider the set E of singularcurves of intersection of C with C = g(C) in M. Modulo Stab(M), there are onlyfinitely many of these. Let T be the topological ball bounded by M and Af. Foreach curve in E there is a subsurface of C lying in T and having that curve onits boundary. These subsurfaces are all disjoint, so we can use separation in theball T to partially order them. One of these subsurfaces, call it Ic, lies closest toM; that is, there is no such subsurface between M and A.

Modulo Stab(N), N has only finitely many boundary loops or pairs ofboundary arcs; let U...... U, be a complete list of inequivalent such loops, wheresome of the U; might be a pair of boundary arcs with the same stabilizer. EachUj bounds a disc Nj on C; let Aj be a parallel disc, lying outside T, where R, is soclose to Nj that they both intersect the same translates of C. As above, we extendor cut off the boundary of A, so that 8A, lies on C. Let A be the union of the A,,j = 1, ..., k, together with all their translates under Stab(1Cl). Replace C by a newsurface C' as follows. Replace N by A, and for every j c- J, replace g o j o g-' (13)by g -j o g-1 (R). Now replace C by C = g-' (C').

Since A lies outside C, all the components of A lie inside C. We chose N sothat it lies closest to C; this means that A does not intersect C, so C' is also anembedded disc. Exactly as above, since IgI is maximal, A cannot intersect anytranslate h(C), where h(W) lies inside C, and if A intersects some translate h(C),where h(W) lies outside C, then there must already be a curve of intersection ofC with h(C).

It is obvious that C and g(C) have fewer curves of intersection modulo J thando C and g(C). Also, for any h e G, C intersects h(C) if and only if C intersects h(C).

B.24. Combining the two steps above, we first fix g of maximal length, and observethat we can successively reduce the number of curves of intersection modulo Jof C with g(C), without introducing intersections of C with any new translateh(C). Hence, after a finite number of steps, we arrive at a new disc C1, where C1does not intersect g(C1), and for every h e G, h(C1) intersects C1 if and only if h(C)intersects C. For each such h, the number of curves of intersection of h(C1) withC1 modulo J might be greater than the number of curves of intersection of Cwith h(C) modulo J, but it is still finite. Hence after a finite number of steps, wewill reach an embedded disc C with no singular curves on it; this is the requiredspanning disc.

VII.C. The First Combination Theorem 149

VII.C. The First Combination Theorem

U. In this section, as in the preceding one, all groups are subgroups of AN, and,unless stated otherwise, are regarded as acting on C. The main results of thissection are stated in C.2; before going on to the statement, we need somedefinitions.

Let { Wm} be a collection of simple loops. We say that the loops { W.)nest about the point x, if each W. separates x from Wm_, (in particular, thismeans that the loops { W. } are all disjoint), and that if zm is any point of Wm, thenzm -x.

C.2. Theorem. Let J be a geometrically finite subgroup of the discrete groups G,and G2. Assume that J # G,, J 0 G2, and that there is a simple closed curve Wdividing t into two closed topological discs B, and B2, where B. is a (J, Gm)-block,and (h, A2) is a proper interactive pair. Let D. be a fundamental set for Gm,where D. is maximal with respect to Bm, D. nB3_m is either empty or hasnon-empty interior, and D, fl W = D2 fl W. Set D = (D, fl B2) U (D2 fl B, ), and setG = <G,, G2>. Then the following statements hold.

(i)G=G1* G2.(ii) G is discrete.(iii) If W is precisely invariant under J in either G, or G2, then, except perhaps

for conjugates of elements of G1 and G2, every element of G is loxodromic.(iv) W is a precisely embedded (J, G)-block, and if B, and B; are both strong,

then so is W.(v) If { W. } is a sequence of distinct G-translates of W, then dia(Wm) - 0.(vi) There is a sequence of distinct G-translates of W nesting about the point

x if and only if x is a limit point of G which is not G-equivalent to a limit point ofeither G, or G2.

(vii) D is a fundamental set for G. If D1 and D2 are both constrained, W meetsaDm in a finite set of points, and there is a constrained fundamental set E for J,containing both D, and D2, so that, except for some excluded cusps, E - D. isbounded away from W, then D is constrained.

(viii) Let Qm be the union of the Gm translates of Am, and let R. be thecomplement of Qm. Then Q(G)/G = (R1 fl Q(G, ))/G, U (R2 fl Q(G2 ))/G2, wherethese two possibly disconnected surfaces are identified along their common possiblydisconnected and possibly empty boundary, (W fl 6(J))/J.

(ix) Assume that W is strong. Let C be a precisely invariant topological disc inH3 spanning W, so that C divides H3 into two closed sets. B; and B2, where Bm isprecisely invariant under J in G. H'/G can be described as follows; it is the unionof H 3/G1, from which the image of Bl /J has been deleted, and H 3/G2, from whichthe image of BZ/J has been deleted, where the two 3-manifolds are joined alongtheir common boundary, C/J.

(x) If Bt and B2 are both strong, then, except perhaps for translates of limitpoints of G, or G2, every limit point of G is a point of approximation.


(xi) G is geometrically finite if and only if G, and G2 are both geometricallyfinite.

Before going on to the proof of this theorem, we remark that conclusion (vii)might be vacuous; i.e., it might be that D = 0.

We also remark that the hypothesis that (E,, 62) be a proper interactive pairis usually easy to check in practice. First of all, since B. is a block, tm is preciselyinvariant under J in Gm; then A.6 asserts that the pair is interactive. The onlyadditional hypothesis we need to check is that it be proper. This means that eitherthe G,-translates of Bt do not cover all of B2 or that the G2-translates of B2 donot cover all of Bt. It is easy to see that this will hold if either D, n B2 # 0, orD2 n B, 0 0.

The remainder of this section is devoted to the proof of this theorem; weassume throughout that the hypotheses hold.

C.3. We saw in A.6 that (a,, h2) is an interactive pair, and we have assumed itto be proper. Conclusion (i) now follows from A.10.

C.4. To prove conclusion (ii), assume that G is not discrete. Then there is asequence (g. } of distinct elements of G, with gk(x) -+ x, uniformly on compactsubsets. By A.9, we can assume that each g, has even length, for if Ig,kl is odd,then either gk(B,) c B2, or gk(B2) c B,.

Assume without loss of generality that the G,-translates of Bt do not coverall of B2. If g = g o o g 1 is a (1,2)-form, then g(B2) c g (B,) c B2. If there isonly one set of the form h(B,) contained in B2, where h e G,, then B2 - g(BI) hasnon-empty interior; if there is more than one such set, then B2 - g (B,) surelyhas non-empty interior. In any case, there is a compact subset of B2 on which gis far from the identity. Similarly, if g is a (2,1)-form, g(B,) c g og c g (B2) g o g1_ I (B,) has non-empty interior, so does B, -g(B, ); we conclude that there is a compact subset of B, on which g is far fromthe identity.

C.S. We next prove conclusion (iii). Let g be some element of G, where IgI isminimal among all conjugates of g in G. If IyI = 1, there is nothing to prove;assume IgI > 1. Write g = og, in normal form. Since IgI is minimal, thelength of g is even. Then g is a (3 - m, m)-form, and g(BB) c g o (B.) c Bm.Since aBm = W is precisely invariant under J in either G, or G2, the last inclusionis not only proper, but g o g,_, (Bm) c Am. Hence g has infinite order, and g hasat least two fixed points; one in g o (Bm), and the other in B3_m. Since g hasinfinite order, it is not elliptic; since it has at least two fixed points, it is notparabolic.

C.6. We next look at the set of G-translates of W. If IgI = 0, then g(W) = W. IfIgI = 1, then either g e G,, in which case,g(W) = g(aB,) c B2, or g e G2, in whichcase g(W) c B1. This set of translates of W divides C into regions; roughly


speaking, we set S, to be the union of those two of these regions having W ontheir boundary. More precisely, we set

Ti.m = U g(B,),g C G,,, -J

set T, = Tt,1 U T,,2, and let S, be the complement of T, in :.Notice that T,,m is contained in B3_,,,. If Tim = B3_m, then there is nothing

further to be said. If there are several distinct translates of B. to B3_m, then theseall have disjoint interiors, but of course, we have no control over how the bound-aries might intersect. In this cases, nB3_m is the common exterior, inB3_m, ofall these sets {g(Br)JgeG. - J}.

Continuing inductively, we set

Tn.m = U g(Te-l.3-,n),

set T. = T,,,, U and let S. be the complement ofNotice that for fixed m, {T,,,m} is a decreasing sequence of sets; that isc T,,,m c . . . c T1,m c B3, Then { T.) is also a decreasing sequence of sets,

and is increasing.In the case that W is precisely invariant under J in both G, and G2, we can

easily describe these sets. Let zo be some point on W. For every z E C, define the"distance" from z to W to be 1 + the minimal number of translates of W a pathconnecting z to zo must cross. Then the closure of S, is the set of points at distance1 from W, and the interior of Tt is the set of points at distance greater than 1from W. Similarly, fi is the set of points at distance greater than n from W, andthe closure of (S - is the set of points at distance precisely n from W.

The description above shows that if W is precisely invariant under J in bothGt and G2, then it is precisely invariant under J in G. It also shows that W isprecisely embedded in G.

C.7. Lemma. Let z be a point of s, f1E3_m. Then either zeIT(Gm) or there is ag e Gm, and there is a point y e D with g(y) = z.

Proof. For every z 0 )T(Gm) there is a point y E D. and a g e Gm, with g(y) = z.Since z e St, z does not lie in any Gm translate of Bm; since D. is maximal, y f Bm.So y e D. fl B3_m c D. 0C.8. Lemma. D c St.

Proof. Let x E D, fl B2; the proof for points of D2 fl Bt is essentially the same.Assume that x e T1; then x lies in a G1-translate of Bt. Since Dt is maximal, x isG1-equivalent to some point of D, fl Bt . This can occur only if x e Dt (1 Bt fl B2 =D,fW

Since Bt is a block, for every g E G, - J, where g(W) fl w $ 0, the points ofintersection must be limit points of G,. This shows that no point of W fl T, liesin Q(G 1). Hence x $ Tt, so x c- St. 0


C.9. Lemma. D c °Q(G).

Proof. We first show that D c Q(G); assume not. Then there is a limit point z ofG in D; assume for definiteness, that z e D, fl B2. Then (D, fl B2) has non-emptyinterior, and is precisely invariant under the identity in G, so G is Kleinian. Sincez E A (G), there is a sequence {g,k} of distinct elements of G, with gt(y) -> z, for allpoints y with at most one exception. Assume first that z does not lie on W. Forany 2-form g, g(B) c B,, where B is either Bt or B2; hence every gk is a 1-form.If Igkl > 1, then for B = B, or B2, gk(B) c T,; hence zetT,. Of course the pointson c0 T, are either G,-translates of points of W, or limit points of G,; we concludethat z is either a point of W, or a translate of such a point. Since D, fl W andD2 fl W are both maximal, z actually lies on W.

Every point of D, fl W lies in Q(G, ); since B, is a block, no G,-translate ofB, passes through z. Since z e Q(G, ), the G,-translates of B, do not accumulateat z. Hence there is a neighborhood of z that is disjoint from all non-trivialG,-translates of W (i.e., z is an interior point of S,). Then the G-translates of Wdo not accumulate at z. Hence z e Q(G).

Since D c Q(G), and D is precisely invariant under the identity in G,D c °Q(G).

C.10. We now prove conclusion (iv). Every point of W fl °Q(J) is J-equivalentto some point of W fl D, = W n D2 = W n D c °Q(G). Hence w fl °Q(J) =W fl °Q(G). Also since D n W is a fundamental set for the action of J on W, andD is precisely invariant under the identity in G, w n °Q(G) is precisely invariantunder J in G.

Since J keeps both B, and B2 invariant, there cannot be any elliptic fixedpoints of J on W. Hence w fl Q(J) = W fl °Q(J) = W fl °Q(G) c w fl Q(G). Theopposite inclusion is trivial. We have shown that W is a (J, G)-block.

We know that W is precisely embedded in both G, and G2. The fact that Wis precisely embedded in G is an immediate consequence of the fact that thesequence of sets is decreasing. It remains to show that if B, and B2 are bothstrong, then so is W.

Let x be a rank I parabolic fixed point of J. Since B. is strong, x is either arank 2 parabolic fixed point of or is doubly cusped in If Stab(x) has rank2 in either G, or G2, then it surely has rank 2 in G, in which case there is nothingto prove. Choose a doubly cusped region C = C, U C2 near x, where C. c B.,and C is precisely invariant under Stab(x) in both G, and G2. Let J. = Staba,,,(x).

It is easy to see that we can make C, sufficiently small so that exactly one ofthe following three possibilities holds. Either

(i) C, cS,,or(ii) there is a g2 a G2 - J, with g2(x) = x, so that g2(C,) is a cusped region for

x in both G, and G2, or(iii) there is a g2 e G2 - J, so that g2(C,) is a cusped region for both G, and

G2, but y2 = g2(x) is not J-equivalent to x.

VILC_ The First Combination Theorem 153

Similarly, we can make C2 sufficiently small so that either(i') C2 c S,, or(ii') there is a g, e G, - J, with g, (x) = x, so that g, (C2) is a cusped region

for both G, and G2, or(iii') there is a g, E G, - J, so that g,(C2) is a cusped region for both G, and

G2 in B,, but y, = g, (x) is not J-equivalent to x.It is easy to see that if (ii) and (ii') both do not hold, then StabG(x) is parabolic

cyclic, and x is doubly cusped in G.If case (ii) holds, then g2(x) = x, so 92 is either parabolic or elliptic. Since E2

is precisely invariant under J in G2, 92 is not parabolic, and 92 can only be ahalf-turn. If case (ii') does not hold, then StabG(x) is not cyclic, but has rank 1.We have the same result if case (ii') holds, but case (ii) does not.

If (ii) and (ii') both hold, then g, and 92 are both half-turns with fixed pointx. The product h = g, o g2 is parabolic with fixed point x. By conclusion (i), h 0 J;in this case, StabG(x) has rank 2.

C.11. Conclusion (v) follows from conclusion (iv), and B.14.

C.12. Let S = U S,,, and let T = fl T. be its complement. The sets { aredecreasing. A point x lies in T if and only if there is a sequence { g,,) of elementsof G with 19k1 -+ 00, X E 92k(B2), and x E92k+1(B1) (or xE92k(B1), and X E92k+1(B2))It is clear that T, and hence also S, is G-invariant.

Proposition. Every point of S is a G-translate of either a point of D, or a point of2T(G1), or a point of 2(G2).

Proof. The sets S are increasing, so if x c- S, there is some index n, so that x E S.,but x 0 1. Then there is a g of length n - 1, so that g(x) E S, . Hence it sufficesto consider only points of S1.

The points of S1 all lie in either B1 or B2. Assume that x e B1; the proof in theother case is essentially the same. Then x lies either in 2(G2) or in °Q(G2). In thelatter case, there is a g E G2 with g(x) * B2. Hence g(x)eB,f1D2cD. O

C.13. The points of A(J) can lie in either S, or T,. If W is precisely invariantunder J in both G1 and G2, then A(J) c S,. If W is not precisely invariant underJ in say G1, then there is a point z on both W and g(W), for some J.This point z, which is necessarily a limit point of J, is then a point of T,, not S.

Except perhaps for some points of 4(J) and their translates, every point ofD U 2T(G1) U ?(G2) 'S contained in St. Since S is G-invariant, every G-translate ofany point of (D U IT (GI) U 2(G2 )) - A(J) lies in S.

C.14. Consider a point z c- T. Assume for simplicity that z c- B2. Since z c- T1, thereis an element h, = g, e G, - J, so that z c- g 1(B1); then since z e T2, there is an

154 V11. Combination Theorems

element g, a G2 - J, so that z e g, o g2(B2) = h2(B2) c h, (B, ); then since z e T3,there is an element g3 a G, - J so that z e g, 092093(BI) = h3(BI) c h2(B2) eh, (B, ); etc. The elements {hk} have increasing length, so the sets {hk(W)} are alldistinct. We have shown that if z e T, then there is a sequence {hk} of elements ofG, with IhkI - oo, and zE c hk( k) c . c h2(R2) c h,(61), where E, is alter-nately equal to B, and B2.

There are two possibilities for the sequence above. Either z lies in the interiorof infinitely many of the hk(,O;), or, from some point on, z lies on the boundaryof every hk(Q,,). In the former case, there is a subsequence {hk} so that hk(W) nestsabout z. In the latter case, z is a translate of a point w of W. Since W is a block,and there are many translates of W intersecting W at w, w is a limit point of J.We have shown that for every point x of T, either there is a sequence of translatesof W nesting about x, or x is a translate of a limit point of J.

Proposition. T c A(G).

Proof. If hk(W) nests about x, then hk(w) - x for all w e W. 0C.15. We now prove conclusion (vi). If x is a limit point of G which is notG-equivalent to a point of either A(G,) or A(G2), then by C.12, xe T. We sawabove that every point of T is either a translate of a limit point of J, or is thelimit of a nested sequence of translates of W.

Now assume that there is a sequence {hk(W)} of distinct translates of Wnesting about z. Then z is a limit point, and is surely not in S. The only possibilityleft is that z lies in T, and is a translate of a limit point of J. Since W is a preciselyembedded block, there cannot be a point of W with a sequence of translates ofW nesting about it.

C.16. The first part of conclusion (vii) is almost immediate. We already knowthat D c °Q(G), and that D is precisely invariant under the identity. We saw inC. 12 that every point of S is a translate of some point of D U A(GI) U A(G2), andwe saw in C.14 that T c A.

Suppose that D, and D2 are constrained, and, except perhaps for someparabolic cusps, D, and D2 both agree with E in a neighborhood of W. Then thesides of D in B, are just the sides of D2 in B,; likewise the sides of D in B2 are thesides of D, in B2. These sides are paired, and they can accumulate only at limitpoints. The only thing left to show is that the tesselation of Q(G) by translatesof D is locally finite.

Suppose there is a sequence {hk} of elements of G, with hk(D) accumulatingat x e D fl Q(G). Since D, is constrained, x is not an interior point of s, fl B2; also,since D2 is constrained, x is not an interior point of S, fl B, . Finally, since D C E,E is constrained, and E - D. is bounded away from Wax does not lie on W.

C.17. Once we have chosen D, and D2 so that D is constrained, conclusion (viii)follows almost at once from conclusion (vii).


C.18. We could equally well have constructed fundamental sets in H3, and soprove conclusion (ix). We first remark that the existence of the spanning disc Cis guaranteed by B.16. Consider the set of translates of C. These divide H' intoregions, where there are exactly two inequivalent such regions. There are exactlytwo of these regions which together span S,; call them E; and E. It is almostimmediate that E' is precisely invariant under J in G..

We conclude from B. 14 that if { g,E(C)} is a sequence of distinct translates of C,then diaE(gk(C)) - 0. It follows that every point of Fl' is G-equivalent to eithera point of E;, or of E2, or of C. Conclusion (ix) now follows.

It should be remarked that we can easily put conclusion (ix) in the sameformat as conclusion (viii); that is, we could state conclusion (ix) as follows. Let

be the complement in H' of the union of the G.-translates of B.. Then H'/Gis the union of E;/G, and EZ/G2, where these two 3-orbifolds are joined alongtheir common boundary, C/J.

C.19. We come now to conclusion (x). Suppose x is a limit point of G, which isnot a translate of a limit point of either G, or G2. Then x e T, there is a sequence{hk} of distinct elements of G, and there is a choice of B = B, or B = B2, so thatx e - hk(B) c c h, (B). Assume for simplicity that B = B1, we also assumewithout loss of generality that h, = 1. Then W separates h,, `(W) from hk t(x).For every k there is an element jk e J so that fk(x) = fk o hk' (x) e E, a constrainedfundamental set for J. Since W is strong, there is a neighborhood U of W withD fl u = E fl U, except perhaps for some exceptional cusps. We first take up thecase that the fk(x) are all bounded away from W; since W separates fk(x) fromfk(W), there is some b > 0 so that I fk(x) - fk(z)I >- 6, for any point z lying on aspanning disc for W (we have renormalized so that G acts on B'). Hence x is apoint of approximation.

Next assume that fk(x) we W; this can occur only if w is a parabolic fixedpoint of J. Since W is strong, and x is a limit point of G, no translate of x canenter a rank 1 exceptional cusp. Hence fk(x) - w in a rank 2 exceptional cusp.Let, j be a primitive parabolic element with fixed point w, where j * J. Notice thatj(W) fl w = w; the powers j2(W) divide C into regions; the sequence of points{ fk(x)}, which tends to w inside the cusp, necessarily passes through infinitelymany of these regions. Hence there is an increasing (or decreasing) sequence ofpowers {ak}, so that j k o fk(W) -+ w, while .j'k o fk(x) is bounded away from w.As above, this implies that for any point z lying on a spaning disc for W,If"o /k (x) -.%°kofk(z)I 2t 6 > 0.

C.20. We start the proof of conclusion (xi) with the assumption that G, and G2are both geometrically finite. Then B, and B2 are both strong: conclusion (iv)then asserts that W is strong.

Let x be a limit point of G. Suppose first that x is a parabolic fixed point;there is nothing to prove except in the case that H = StabG(x) has rank 1. Theparabolic subgroup Ho of H is a translate of a subgroup of either G, or G2, we


can assume without loss of generality that it lies in G,. If x lies on W, orequivalently on some translate of W, then since W is strong, x is doubly cuspedin G. If x does not lie on any translate of W, then we can find a doublycusped region C for Ho, where C does not intersect any G,-translate of W.Then C B2 fl °Q(G, ), so every point of C is G,-equivalent to some point ofD, fl B2 c D c °Q(G). By VI.E.5, x is doubly cusped in G.

Next assume that x is a limit point of G, which is not a parabolic fixed point.If x is a translate of a limit point of G,, then x is a point of approximation forG,, and hence for G. Similarly for G2. If x is not a translate of a limit point ofeither G, or G2, then by conclusion (x), it is a point of approximation.

C.21. Finally, we take up the converse; assume that G is geometrically finite, andthat G, is not. If x is a rank I parabolic subgroup of G, which is not doublycusped in G then it is surely not doubly cusped in G, so StabG(x) must haverank 2. This can occur only if there is a parabolic element of G - G, commutingwith the parabolic element of G,; this in turn can occur only if x lies on sometranslate of W; it suffices to assume that x c- W. Since there is a parabolic elementof G - G, with fixed point x, there are infinitely many translates of W passingthrough x and lying in B,, similarly there are infinitely many translates of Wpassing through x and lying in B2. Then there must be an element g. e G., - Jwith x E g.(W) fl W. In particular, there is a g, a G, - J so that x lies in g, (B, ).Since 1), is precisely invariant under J in G there is a cusped region C, for xinside B, , and there is a cusped region C2 for x inside g, (B,). Then C, U g, (C,)is a doubly cusped region, precisely invariant under StabG,(x). We have shownthat if Stab6(x) has rank 2, then x is necessarily doubly cusped in G,.

The only other possibility is that there is a limit point x of G, which is not aparabolic fixed point, and is a point of approximation for G but not for G,. Thenx lies in the closure of B2 fl S,. Since every limit point on W is also a limit pointof J, and J is geometrically finite, we can assume that x does not lie on anytranslate of W. Let {hk } be the sequence of elements of G with d(hk(x),hk(z))bounded from below for almost all z (here d(-, ) denotes spherical distance); sincex is not a limit point of J, we can assume that the hk represent distinct left Jcosets; that is, the sets {hk(W)} are all distinct. If IhkI = n, and hk is an (i,j}formwhere j = 1, then hk(S, fl B2) c while if j = 2, hk(S, fl B2) c T. Of course,int(hk(S, fl B2)) is contained in exactly one connected component of t-, or

Since dia(hk(W)) - 0, d(hk(x), hk(z)) 0 for all z e S, fl B2, contradicting theassumption that d(hk(x), hk(z)) is bounded away from zero for all z, with at mostone exception. 0

VII.D. Combinatorial Group Theory - II

D.I. Our basic hypothesis in this section is that we are given two groups, Go andG,, and we are given two subgroups J, and J2 of Go, so that the following hold.

VII.D. Combinatorial Group Theory - II 157

Go and G, have trivial intersection; G, is infinite cyclic, with generator f; notonly are J, and J2 abstractly isomomorphic, but f conjugates J, into J2; that is,j _- f o j of -' defines an isomorphism f*: J, -- J2. We make no assumptionsabout J, fl J2 or about the index of J. in Go.

D.2. A normal form is a word of the form f '-g.... f", g 1, where each gk E Go; fork > 1, gk 0 1; the ak are integers, with only at. permitted to be 0; if ak < 0 and91,+1 E J, - (1), then ak+, < 0; if ak > 0, and gk+1 E J2 - 11), then ak+, > 0.

Two normal forms are equivalent if we can get from one to the other by afinite sequence of insertions and deletions of words which we know to be theidentity; i.e., conjugates and inverses of words of the form, fjf -' (f*(j))-'.

It is clear that every word of the form f°'g1 is equivalent eitherto a normal form or to the identity. The set of normal forms modulo equiva-lence, together with the identity, forms a group, where the operation is juxta-position of words. This group is called the HNN-extension of Go by f, and iswritten as Go *f .

Note that Go * f cannot be trivial, for it contains G, = <f >, which is of infiniteorder.

D3. There is a natural homomorphism 0: Go* f - <Go, G, >, where 0 re-places juxtaposition by composition of mappings; that is, 0(f a°g,, ... f °' 91) =fan o g o o f=i o g 1. It is obvious that equivalent normal forms are mapped ontothe same transformation.

As with amalgamated free products, every element of Go* f has many differentequivalent normal forms. One easily proves that every normal form is unique ifand only if J1 = J, = l; i.e., G is the free product of G1 and G2.

If 0 is an isomorphism, then we say that G = <Go, G1 > = Go* f. It is easy tosee that 0 is an isomorphism if and only if no nontrivial normal form lies in thekernel of 0. If 0 is an isomorphism, then we will regard <Go, G1 > as being Go *f ,

and we will write normal forms using composition (a o b) rather than juxta-position (ab).

DA. Every normal form g = f a^g,,... f °'g1 has a length, defined as I g I = Y Ia.1.Note that equivalent normal forms have the same length, so IgI is well definedfor every element of Go *f .

D.S. The normal form f '-g,... f g, is positive if a > 0, negative if a < 0, andis a null form if a = 0. For a null form, note that if > 0, then gp 0 J2, whileifa.-1 <0,9n0J1

If g is a positive, respectively negative, respectively null, normal form, thenevery equivalent normal form is positive, respectively negative, respectively null.Hence it makes sense to talk about an element of Go * f as being positive, negative,or null. We write g > 0, g < 0, or g - 0, for these three cases.


D.6. Using normal forms, it is easy to see that isomorphisms can be extended.That is, suppose we are given two HNN-extensions G = Go*f, and 6i = 0o*7,where f conjugates J, into J2, and f conjugates I, into .72. Suppose we arealso given isomorphisms tpo: Go - CO, and q1: <f> -> <f>, where V1(f) =f, (po(Jm) = 3m, and cpo o f* I J, = f* o rpoIJ1. Then these isomorphisms can beextended to an isomorphism (p: G - 6.

D.7. We return to our original assumptions on G = <Go, f>. An interactive tripleis a triple of disjoint nonempty sets, (Z, X1, X2) satisfying the following: (X1, X2)is precisely invariant under (J,, J2) in Go; for every g e Go, g(Xm) c (Z U Xm);f(ZUX2) C X2;f-1(ZUX1) C X1.

We specifically mention one consequence of these conditions. If (Z, X1, X2)is an interactive triple, and g e Go - J, , then g(X1) a Z; similarly, if g e Go - J2,then g(X2) (-- Z.

D.S. Theorem. If G acts freely and discontinuously on (some non-empty opensubset of) X, and G = Go*fs then there is an interactive triple of sets.

Proof. Let D be a fundamental set for G. If Go 96 { 1), set

Z = U g(D), X1 = U g(D), X2 = U g(D).9_0 9<0 g>O

Since D 0 0, Z 0. Since G, is infinite cyclic, X, and X2 are not empty. Itis almost immediate that if g > 0, and h e Go, then hog is never negative, and ispositive if and only if h e J2. Similarly if g < 0, then hog is never positive, and isnegative if and only if h e J,. This shows that for every h e Go, h(Xm) c Z U Xm.It also follows that (X1,X2) is precisely invariant under (J,,J2) in Go.

Also, if x e D, and g(x) a Z, then for a > 0, f a o g(x) lies in X2, while for a < 0,it lies in X1 . if g(x) a X2 (i.e., g > 0), and a > 0, then f" o g > 0, so P0 g(x) a X2;similarly, if g(x) E X 1 (i.e., g < 0), and a < 0, then fl o g < 0, so f a o g(x) e X 1.

If Go = { 11, then choose some point x, let Z = D = {x}, and proceed as above.0

D.9. The converse to D.8 is false, but there is no easy counter-example.

D.10. For the next application, we need the notion of an (j, k)-form, wherej = "+", "-", or "0", according as g is positive, negative, or null, respectively;k="+" if a, >0,k="-" if a, < 0.

Assume we are given an interactive triple (Z, Xt, X2); let Zo be the set of pointsin Z that are not Go-equivalent to any point of either X, or X2. Observe that Zois Go-invariant.

VII.D. Combinatonal Group Theory - II 159

D.11. Lemma. Let g = f'-g,,... f °'g, be a normal form in G, where IgI > 0.(i) If g is a (+, +)-form, then P(g)(Zo U X2) c X2.(ii) If g is a (+, -)-form, then b(g)(Z0 U X,) c X2.(iii) If g is a (-, +)-form, then '(g)(Zo U X2) c X,.(iv) If g is a (-, -)-form, then o(g)(Z0 U X,) c X,.(v) If g is a (0, +)-form, then there is an element he Go, so that 0(g)(Zo U X2) c

h(B)cZ,where B=X, if <0,and B=X2ifa._, >0.(vi) If g is a (0, -)-form, then there is an element h e Go, so that P(g)(Zo U X,) e

h(B) c Z, where B = X, if a.-, <0,and B=X2if >0.

Proof. The proof is by induction on Jgi. We start with the case that JgJ = 1, andwrite g = g2fg,, or g = g2f -'g,; for simplicity, assume the former. Since Zo isGo-invariant, and 9, (X2) C Z U X2, g, (Zo U X2) c Zo U Z U X2 = Z U X2. Thenf-91 (Z0 U X2) c X2, which, for g2 = 1, is conclusion (i). If g2 # 1, then 92 tl J2,so 92(X2) c Z; i.e., g2 o f og,(Z0 U X2) c gx(X2) c Z; this is conclusion (v).If g is of the form 920f_'091, then similar arguments yield conclusions (iv)and (vi).

Now assume that fig( > 1, and that our conclusions hold for all normal formswith length less than Jgj. Write g = f g,, and assume for simplicity thatat, > 0; again, the proof in the other case is essentially the same.

We first take up the case that g is a (+, +)-form. If a > 1, then f-'g is a(+, +)-form of length 191 - 1, so P(f -' g)(Z0 U X2) C X2. Apply f to obtain0(g)(Zo U X2) c f(X2) C X2.

If a,, = 1, then f ` g need not be a normal form. If f -'g is a normalform, then it is a (0, +)-form, so b(f -' g)(Zo U X2) c Z. Then since f(Z) c X2,(ft(g)(Z0 U X2) C X2. If f -'g is not normal, then since a > 0, it cannot be that

eJ,, and 0; the only other possibility is that 6J2, and > 0.In this case, g-'1 f -'g is a (+, +)-form, so f -' g) maps Zo U X2 into X2;then which is in J2, maps X2 onto itself; finally, f maps X2 into X2.

We next take up the case that g is a (-, +)-form, and proceed as above. Ifa < -1, then fg is a (-, +)-form of length I gI - 1, so P(fg)(Zo U X2) C X1.Then f1 o f o 0(g)(Zo U X2) c f -' (X,) c X,. If a = -1, and fg is a normalform, then fo0(g)(ZoUX2)cZ, and f-'ofoP(g)(ZoUX2)c f-'(Z)cX,.If fg is not normal, then eJ,, and < 0; in this case, g-_',fg is a(-, +)-form of length IgI - 1, so ch(g;_',fg) maps ZQUX2 into X,. Since9._,eJ fob(9)(Z0UX2)-9.-1(X0=X, Then f-'ofo.b(9)(Z0UX2)cf-'(X1) c X1.

We finally take up the case that g is a (0, +)-form. Then g,-'g is normal andis either a (+, +)-form or a (-, +)-form of length IgJ; we have just seen that if

0, then 0(g-'g)(Zo U X2) c X2, while if 0, then U X2) cX,. In the former case, since g is normal, g,, O J2, so g(X2) c Z; hence(g)(Z0 U X2) c c Z. Similarly, if 0, then o U X2) eX and since g 11 J g(Z0 U X2) c c Z. 0


D.12. As in A.10, the converse of D.8 requires an additional topological assump-tion. Recall that Zo is the complement of the set of translates of X1 U X2 in Z.The interactive triple is proper if Zo 0 0.

Theorem. Suppose there is a proper interactive triple of sets (Z, X,, X2) forG = <Go, f ). Then G = Go *t,

Proof. Let x be a point of Zo, and let g be some normal form of positive length.By D.11, 0(g)(x) x. Hence 0(g) 0 1. Since no normal form is mapped to theidentity in G, G = Go * f.

D.13. A fundamental set Do for Go is maximal with respect to the interactive triple(Z, X1, X2), if Do fl x. is a fundamental set for the action of Jm on X..

Theorem. Assume that Go acts discontinuously on some open subset of X, and that(Z, X1, X2) is an interactive triple. Let Do be a maximal fundamental set for Go.Then Zo is precisely invariant under Go in G, and D = Do fl z is precisely invariantunder the identity in G.

Proof. Since Do is maximal, Do fl Z = D fl Zo. We saw in D.11 that for every gof positive length, o(g)(Z0)flZo = 0. Of course, if Igi = 0, then gc- Go, andg(D) n D = 0.

VILE. The Second Combination Theorem

E.I. As in VII.C, all groups in this section are subgroups of M, and are regardedas acting on C (again, the necessary modifications for the case that G c FA areleft to the reader). The main results of this section are gathered together in thestatement below. We assume throughout that J, and J2 are geometrically finitesubgroups of the discrete group Go, and that G, is infinite cyclic with generator f.

Two closed discs B, and B. are jointly f-blocked if B. is a (.1,,, Go)-block,(B, n Q(G0), B2 n Q(Go)) is precisely invariant under (J,, J2) in Go, f maps theexterior of B, onto the interior of B2, and fJ, f -' = J2. If B, and B2 are jointlyf-blocked, then let A be the common exterior of B, and B2, and let A0 be the setof points in A that are not Go-equivalent to any point of either B, or B2-

A fundamental set Do for Go is maximal if Do n B,, is a fundamental set for theaction of J. on B,,, and if f(Do n W,) = Do n W2, where W. = aBm. Notice thatif B, and B2 are jointly f-blocked, then a maximal fundamental set necessarilyexists.

E.2. We need some observations about jointly f-blocked sets. The first is thatsince (B, n 'Q, B2 n Q) is precisely invariant under (J,, J2), B, and B2 have

VILE. The Second Combination Theorem 161

disjoint projections to f2/Go, from which it follows that W, and W2 are bothprecisely embedded in Go; that is, W. divides C into two open discs, and everyGo-translate of either W, or W2 does not intersect one of these two discs. Thesecond observation is that since Bis a block, Em c Q(G0). Also note that sincef conjugates J, onto J2, it maps parabolic fixed points of J, onto parabolic fixedpoints of J2; of course, the parabolic fixed points of J. lie on W. = aBm.

We state the next two observations as separate propositions. The first followsalmost immediately from the definitions.

E.3. Proposition. If (B1,B2) is precisely invariant under (J,,J2) in Go, and f mapsthe outside of W, onto the inside of W2, conjugating J, onto J2, then B, and B2 arejointly f-blocked.

E.4. Proposition. If B, and B2 are jointly f-blocked closed topological discs, andA is the common exterior of Bt and B2, then (A, E1, A2) is an interactive triple.

Proof. Since (A,, A2) is precisely invariant under (J, , J2), every element of Go maps.6, either onto A, or into A; similarly with B2. The other statements in thedefinition of interactive triple are immediate.

E.5. Theorem. Let J, and J2 be geometrically finite subgroups of the discrete groupGo, and let G, = <f > be infinite cyclic. Assume that B, and B2 are jointly f-blockedclosed topological discs, and that Ao # 0. Let Do be a maximal fundamental setfor Go. Set G = (Go, f >, and set D = Do fl (A U Wt), where W. = cBm. Then thefollowing statements hold.

(i) G = GO*f.(ii) G is discrete.(iii) If (B1,B2) is precisely invariant under (J1,J2) in Go, then every non-

loxodromic element of G is conjugate to an element of Go.(iv) W, is a precisely embedded (J1, G)-block; if B, and B2 are both strong

Go-blocks, then W1 is a strong G-block.(v) If {W,,') is a sequence of distinct G-translates of W1, then dia(W,t) - 0.(vi) There is a sequence of distinct translates of W, nesting about the point x

if and only if x is a limit point of G, and x is not a translate of a limit point of Go.(vii) If B, and B2 are both strong, and x is a limit point of G which is not

G-equivalent to a limit point of Go, then x is a point of approximation.(viii) D is a fundamental set for G. If Do is constrained, W, and W2 intersect

3Do in a finite set of points, and if there is a constrained fundamental set E. forJ. so that, except perhaps for some excluded cusps, Do and E. agree near Wm, thenD is constrained.

(ix) Ao is precisely invariant under Go in G. Let Q = Ao fl D(GO); thenQ(G)/G = Q/Go, where the two possibly disconnected and possibly empty bound-aries, (W1 fl Q(Go))/J, = (W, fl Q(J1))/J1 and (W2 f1 Q(Go))/J2 = (w2 n Q(J2))/J2are identified; the identification is that given by f (that is, if xe W1, then p(x) isidentified with p o f(x)).

162 VI!. Combination Theorems

(x) G is geometrically finite if and only if Go is geometrically finite.(xi) Assume that W, and W2 are both strong. Let C. be a spanning disc for W.

where C2 = f(C, ). Let B.,, be the topological half-space cut out of H3 by Cm, whereB,3, spans Bm. Then 0-03/G can be realized as H3/Go, where the images of Bl /J, andB2/J2 have been deleted, and the two resulting boundaries, C,IJ, and C21J2, areidentified; the identification being given by f.

Before going on to the proof of this theorem, we remark that conclusion (viii)may be vacuous; that is, D might be empty.

The condition that AO 0 0 is not transparent. Notice that if x is a limit pointof Go which is not a translate of a limit point of either J, or J2, then x e Ao. Also,there will be regular points in Ao if D # 0; that is, if the projection of B, U B2 isnot dense in Q/Go.

E.6. Conclusion (i) follows from E.4 and D.12 (the statement that Ao is not emptyimplies that the interactive triple (A,,6,, h2) is proper).

E.7. To prove conclusion (ii), suppose there is a sequence of elements {gk}eGwith g,, - 1. We consider each g,1 to be a normal form, assume first that eachIgkl >- 1, and look at D.11 for the different possibilities. If g is either positive ornegative, then g(Ao) c (B, U B2); hence g,, is either a (0, +) or a (0, -)-form. Butif g is a (0, +)-form, g(162) c A, while if g is a (0, -)-form g($,) c A. The onlyremaining possibility is that each gm a Go, which we know to be discrete.

E.8. Let To.. be the union of all the Go-translates of Bm, and let To = To,, U To,2;let So (= A0) be the complement of To.

More generally, let T,,,, = U g(B,), where the union is taken over all formsof length n, where either g, J, or a, < 0. Similarly, let U g(B2), wherethe union is taken over all forms of length n, where either g, O J2 or a, > 0. LetT. = T,,,, U T,,,2, and let S. be the complement of T,,.

Let g = f =^ o . o g, be a normal form of length n. If either a, < 0, or g1 0 J, ,then g(W,) c T,,. If g, eJ1, and a, > 0, then set g' = gogi' of -'. Write g' _f2 o. o f -1. If a, > 1, theng'(W2) c g'(W2) a T.-,. If a, = 1,then either 92 0 J2, or a2 > 0, so g'(W2) c hence g(W,) c We haveshown that every G-translate of W, lies in some T,,.

In the case that (W,, W2) is precisely invariant under (J,,J2), we can easilydescribe these sets. Let zo be some point in A0. For any point z, define Ilzll to bethe minimal number of G-translates of W, that a path from zo to z must cross.Then S = {zJ Ilzll 5 n}, and t = {zl IlzII > n}.

Lemma. T. c

Proof. Let g be an element of length n > 0, where a, < 0. If g, #J,, then setg' = gogi t o f, and observe that f -1 og,(B,) c B1, Ig'I = n - 1, and g(B,) _


g' o f -' o g, (B,) c g'(B, ). If g, cJ, , then g, (B,) = B,, so we can assume withoutloss of generality that g, = 1 . Set g' = g o f ; if a, < -1, then f 'I" (B,)f" (BI), so 9'(B,) D g(B,), and g'(B,) c If a, = -1, and g2#J1, then./ -1,andg2EJ1,thena2 < 0, s0 f«"o...092(Bt) = f'"o...o.f'2(Bt) D 9(Bt)

The various cases where a, > 0 are treated analogously.

The T. are decreasing, let T be the intersection of all the and let S be itscomplement; i.e., S = U S. We remark that the fixed point(s) of f lie in T.

E.9. We now prove conclusion (iii). Let g = f'" o g. o of ", o g, be a normalform, where g is not conjugate to any element of Go. Using conjugation, andreplacing g by g' if necessary, we can assume that g > 0. Since B, fl B2 = 0,and f maps the outside of B, onto the inside of B2, f is loxodromic. From hereon we can assume that g, # 1. Since a > 0, and IgI is minimal, either at, > 0, or91 9' J2.

If a, > 0, then by D.I l(i), g(B2) c B2. Since (B1,B2) is precisely invariantunder (J,, J2) in Go, and the T. are decreasing, no G-translate of either W, or W2intersects either W, or W2, except of course for the translates under J. Henceg(B2) c E2, so g is loxodromic.

If a, < 0, then g, 0 J2. Then g, (B2) c A, and f" o g, (B2) c B, . Continue asin D. II to observe that g(B2) c B2. Exactly as above, this implies that g(B2) c h2,so g is loxodromic.

E.10. Lemma. W, fl Q(J,) is precisely invariant under J, in G.

Proof. We know that (W, fl Q(J, ), W2 fl Q(J2)) is precisely invariant under (J,, J2)in Go. Hence the Go translates of W, and W2 can intersect only at limit points.Since f maps the limit points of J, onto those of J2, the translates of W, or W2,under elements of length one, also can intersect W, only at limit points of J,. Theresult now follows from E.8.

Ell. Lemma. D c °Q(G).

Let z be a point of D fl A. Since Do is maximal, z does not lie in any translateof either B, or B2. Hence if z were to lie in the closure of the Go-translates ofB, U B2, there would be a sequence of distinct such translates converging to z.Since B, and B2 are both Go-blocks, the diameter of this sequence of translatesof B, or B2 would converge to zero, so z would be a limit point of Go. Wehave shown that every point of D fl A lies in the exterior of the union of theGo-translates of B, U B2; i.e., D n A c A0.

By D.13, AO is precisely invariant under Go in G. Hence every interior pointof AO fl °Q(G0) is also in °Q(G). We conclude that D fl A c °Q(G).


Suppose next that z e D fl W, . Then z and f(z) both lie in Q(Go). Also, sinceB, and B2 are jointly f-blocked, z does not lie on any Go-translate of either B,or B2, other than B, itself. Since z e Q(G0), the Go-translates of B1, other than B,itself, are bounded away from z; Also, the Go-translates of B2 are bounded awayfrom z; i.e., z is an interior point of S, . Now if there were a point x E Q(G), anda sequence of elements {gm} of G, withgm(x) -+ z for all x, with at most one exception; in particular, gm(x) -+ z for somex e B2, which cannot happen, for the translates of B2 all lie in some T,,. We haveshown that no point of D fl w, is a limit point of G. It follows from E.10, that nopoint of D fl w, can be an elliptic fixed point.

E.12. We now prove conclusion (iv). We first show that W, is a block. If Q(J,) flW, = 0, there is nothing to prove. If Q(J,) fl w, # 0, then since W, is a (J,, Go)-block, Do fl w, is a fundamental set for the action of J, on W1. We saw abovethat Do fl w, c °Q(G). It follows that W, fl °Q(G) =w, fl °Q(J, ). Since At is pre-cisely invariant under J,, J, has no elliptic fixed points on W,; i.e., Q(J,) fl w, _°Q(J,) fl w, = °Q(G) fl w, c Q(G) fl W, . The opposite inclusion, Q(G) fl w,Q(J,) n w, is trivial. We now conclude from E. 10 that W, is a (J, G)-block.

We saw in E.2 that W, and W2 are both precisely embedded in Go, and thatno translate of W2 crosses W,. The description of T. in E.8 shows that W, isprecisely embedded in G.

Now assume that B, and B2 are both strong; let x be a parabolic fixed pointon Wt, where StabG(x) has rank 1. Then x and f(x) are both doubly cusped inGo. Let C = C, U C2 be a doubly cusped region at x, where C, lies inside B,.Make C smaller if necessary so that f(C) is a doubly cusped region for Go at f(x).If C2 and f(CI) both lie in So, then, since So is precisely invariant under Go in G,x is doubly cusped in G. The only other possibilities are that there is a g16 Gowith f (CI) c g, (B,), or f (CI) c g, (B2 ), or there is a 92 a Go with C2 c 92(B,), orC2 c g2(B2). In each of these cases, there is a geG with g(W,) # W1, andxeW1flg(W1)

Consider the set of G-translates of W, passing through x; assume that thereare infinitely many distinct such translates; call them {gk(W,)}. Since x is aparabolic fixed point, and every limit point of J, is either a parabolic fixed pointor a point of approximation, but not both, gk 1(x) is also a parabolic fixed pointof J, . Since J, is geometrically finite, there are only finitely many J,-equivalenceclasses of parabolic fixed points. Hence there is a parabolic fixed point y on W,,so that, after passing to a subsequence, gk(y) = x for all k. Then hi = gi 1 ogk hasa fixed point at y, and does not stabilize W,. Since StabG(y) has rank 1, each hkis either parabolic or a half-turn. If hk is parabolic, then StabG(y) has rank 2, soStabG(x) has rank 2. If each hk is a half-turn, then since hk(W,) 0 hj(W,) fork # j,hk o hj is parabolic, and does not stabilize IV,; in this case again, StabG(x) hasrank 2. We have shown that the number of distinct G-translates of W, passingthrough x is finite.

Since no translate of W, can cross W1, the set {gk(W,)} is naturally ordered,


there is an innermost translate (which is either equal to W1, or lies in B1) and anoutermost translate. Make C sufficiently small so that no gk(W,) crosses it. Thenthe inverse of the transformation mapping W, to the innermost, (outermost)translate, maps C, (C2) either into So, or to a set in Bt whose image under f liesin So, or to a set in B2 whose image under f -' lies in So. The result now follows.

E.13. We remark that as part of the proof above, we have shown that if B, andB. are both strong, and if (B, , B2) is precisely invariant under (J, , J2) in Go, thenW, has no exceptional cusps.

E.14. Conclusion (v) follows from the above and B.14.

E.15. Lemma. A(G) fl So c A(G0).

Proof. If x e So is a limit point of G, then there is a sequence {gk } of elements ofG so that gk(z) - x for all z, with at most one exception; in particular, gk(B,) -4 x.Using the nested property of the T;, we see that there is a sequence {hk} ofelements of Go with hk(B, U B2) - x; i.e., x e A (Go).

E.16. Lemma. Every point of T. - T,,. is G-equivalent to some point of SO U W,.

Proof. Let x e T - T.,,, where x is not G-equivalent to any point of Wt. Theneither x e g(E1), where g is a normal form of length n, with either g, 0 J, or a, < 0;or xeg(162), where g is a normal form of length n with either g10J2 or a, > 0.Assume for the sake of argument that g-' (x) a a,. Since x tf Ti+1, fog-' (x) doesnot lie in either B1 or B2. Hence f o g-' (x) e So.

E.17. We now prove conclusion (vi). We start with the assumption that x is alimit point of G, but not a translate of a limit point of Go. By E. 15, x is not atranslate of any point of So. Since W1 is a block, the limit points of G on W1 areall limit points of J1 c Ga; hence no translate of x lies on W1. Since x e To, byE.16, x lies in every T,,; i.e., x e T

Since x e T, there is a sequence of distinct elements {g,) of G, so that for everyk, x e gk(B), where either B = B1 or B = B2. Thus either gk(8B) nests about x, orx lies on the boundary of almost every gk(B). The latter case cannot occur, for ifit did, x would be G-equivalent to a point on W1, necessarily a limit point ofJ, c G.

For the converse, let x be some limit point of Go. Since 6, and E2, and alltheir Go-translates, contain no limit points of Go, x e So. The G-translates of Ware all in To, so no sequence of them can nest about x. If there is an h e G, so thatgk(W1) nests about h(x), then h-' o gk(W1) nests about x, which cannot be.

E.18. For conclusion (vii), let x be a limit point of G, where x is not a translateof a limit point of Go; then there is a sequence {gk} of distinct elements of G, sothat gk(W1) nests about x. We can assume that g, = 1. Then gk' (x) and gk' (W1)


lie on opposite sides of W, , and gk' (x) 0 W, . Find an element jk a J, , so thathk(x) = jk o gk' (x) a E,, a constrained fundamental set for J1, where, except forsome exceptional cusps, E, - Do is bounded away from W,. If hk(x) - W,, thenit does so inside an excluded cusp C, with vertex y.

The cusp C is not a rank I exceptional cusp, for such a cusp contains no limitpoints of G. If C is a rank 2 exceptional cusp, then there is a parabolic elementh e G with fixed point at y, where h 0 J,. One easily sees that if hk(x) - y insideC, then an appropriate power ak of h will map hk(x) away from y, but still on thesame side of W, as hk(x), while mapping hk(W,) closer to y, and still on the otherside of W, . Then hak o hk(x) is bounded away from W,, so hak o hk(x) is boundedaway from hak o hk(W, ). Since hak o hk(x) is bounded away from hak o hk(z) for anypoint z on a spanning disc for W,, x is a point of approximation.

E.19. We saw in E.16 that every point of T. - is G-equivalent to either apoint of S. or a point of W,. Of course, every point of So either lies in A(G0), oris Go-equivalent to a point of D fl A0. Since the points of T are all limit points,we have shown that every point of °Q(G) is G-equivalent to some point of D.

We showed in D. 13 that D fl A0 is precisely invariant under the identity in G.It is an exercise to conclude from the fact that W, is a block that D fl W, isprecisely invariant under the identity in G. This finishes the proof that D is afundamental set for G.

If Do is constrained, then the sides of D are just the sides of Do, together withthe finite set of arcs of W, n b and W2 fl D. It is clear that the sides of D are paired;the sides inside So are paired by elements of Go, and the sides on W, are pairedwith those on W2 by f. It is also clear that if D has infinitely many sides, thenany sequence of these can accumulate only at limit points; in fact, limit points ofGo. Since the tesselation of °Q(G0) by translates of Do is locally finite, there areonly finitely many translates of D in a neighborhood of any point of OD fl so.

If x is a point of aD fl w, then x is also a point of aE fl w,; since J, isgeometrically finite, x e Q(J,) fl w, Q(G0) fl w, Since B, and B2 are jointlyf-blocked, there is a neighborhood U of x, so that U meets no Go-translate ofeither B, or B2, other than B, itself, and f(U) meets no Go-translate of either B,or B2, other than B2. It follows that only the Go-translates of D, together withtheir images under f -', can intersect U. Hence only finitely many G-translatesof D can intersect U. Similar remarks hold if x e aD fl W2. This concludes theproof of (viii).

E.20. Conclusion (ix) follows from conclusion (viii), together with the fact that fmaps w, fl °Q(Go) onto W2 fl °Q(Go).

E.21. To prove conclusion (x), first assume that Go is geometrically finite; thenB, and B2 are both strong. We know from conclusion (vii) that every limit pointof G is either a translate of a limit point of Go, or a point of approximation. Henceit suffices to show that every limit point of Go is either a point of approximation


in G, or a rank 2 parabolic fixed point in G, or a doubly cusped rank I parabolicfixed point in G. Since points of approximation and rank 2 parabolic fixed pointsretain their essence in going from a subgroup to the full group, it suffices toconsider only rank 1 parabolic fixed points of Go. Every such point is of coursedoubly cusped in Go, and is either in So, or is G-conjugate to a point on W,.

If x e So, then there is a doubly cusped region for x in Go lying entirely in So.Since So is precisely invariant under Go, x is doubly cusped in G. Since Go isgeometrically finite, we know from conclusion (iv) that W, is strong; in particular,every rank I parabolic fixed point on W, is doubly cusped in G.

E.22. We assume next that G is geometrically finite, and show that Go is alsogeometrically finite. There are various cases to consider. First, assume that thereis a limit point x of Go, where x is not a parabolic fixed point, and x is not apoint of approximation for Go. Since J, and J2 are both geometrically finite, wecan assume without loss of generality that x E A0. Then x is a point of approxima-tion for G, so there is a sequence (g, } of distinct elements of G, with gk(z) - z'for all z # x, and gk(x) - x' # z'. Since xEA0, by D.11, if IgkI > 1, gk(Ao) iscontained in some translate of either B, or B2. If Igkl - oo, then gk(AO) iscontained in a decreasing sequence of translates of either B, or B2, and thediameter of these translates tends to zero; hence this case does not occur (sinceAO # 0, W, # W2, so there must be infinitely many points between W, and W2;the images of all these points are contained in the decreasing sequence oftranslates of B, and/or B2). If Igkl is bounded, and Igkl > 1, it is still true thatgk(AO) is contained in some G-translate of either B, or B2, which in turn iscontained in some Go-translate of either B, or B2. If these smallest translates ofB, or B2 are all different, then as above, the diameter goes to zero. If they are allthe same, then there is a single element g e G so that g-' o gk(AO) = Ao. Theng-1 o gk a Go, so x is a point of approximation for Go. We have shown that everylimit point of Go is either a parabolic fixed point, or a point of approximation.

If x is a rank 1 parabolic fixed point in Go, then either it is doubly cusped inG, in which case it is surely doubly cusped in Go, or it has rank 2 in G. We nowassume the latter. Since Ao is precisely invariant under Go in G, x 0 Ao, so we canassume that x e W, . The only way we can have x as a rank 2 parabolic fixed pointin G is if there is a Go-translate of either B, or B2 which is "tangent" to B, at x.Since h, and h2 are both contained in Q(G0), there is a cusp at x inside B,, andthere is another cusp at x inside this translate. Hence x is doubly cusped in Go.

E.23. Finally, we prove conclusion (xi). The existence of a spanning disc C, isgiven by (iv) and B.16. Let E be the region cut out of 0-03 by C, and all itsG-translates, where C, and C2 both lie on E. It is clear that E spans So; sinceSo is precisely invariant under Go in G, so is E. It is also clear that E/Go is, exceptfor the identification of C,/J, with C2/J2, the space described in conclusion (xi).Since every sequence of distinct translates of C has (Euclidean) diameter (in B3)tending to zero, every point of 033 is G-equivalent to some point of E. 0


VII.F. Exercises

F.I. Let G be an abstract group; regard G as a topological group with the discretetopology (i.e., every set is open). Let J be a subgroup of G. Then J acts on G byleft multiplication. Show that J acts discontinuously on all of G, and that afundamental set for the this action is a set of right coset representatives.

F.2. Let G, and G2 be subgroups of the abstract group G, and let J be a commonsubgroup of G, and G2. Find necessary and sufficient conditions, in terms of right(or left) cosets, for G to be equal to G, *, G2.

F.3. Let Go be a subgroup of the abstract group G, let J, and J2 be subgroups ofGo, and let f be an element of G - Go, where fJ, f -' = J2. Find necessary andsufficient conditions, in terms of right (or left) cosets, for G to be equal to Go#f.

F.4. Let B be a (J, G)-block satisfying (ii') of B.4, where G is Kleinian, and theprojection of B to S = Q(G)/G is not all of S. Let G = Y gkJ be a coset decomposi-tion. Prove that >dia2(g,E(B)) < oo.

FS. Let B be a (J, G)-block, where G is Kleinian, and the projection of B toS = Q(G)/G is not all of S. Then Y dia°(gk(B)) < co, where G = > gkJ is a cosetdecomposition.

F.6. A discrete subgroup G of hA is of the first kind if A (G) is all of S2; it is of thesecond kind otherwise. Let J be a geometrically finite subgroup of the discretegroup G c fy0, where J is of the first kind. Show that [G : J] < oo.

F.7. Let W be as in C.2, but suppose that W is not precisely invariant underJ in either G, or G. Suppose further that there are elements fk e G, - Jand gk E G2 - J, and there is a point x c- W, so that g o of, (x) = x. Theng = ga o ... of, is parabolic.

F.S. Let C,, C;, ... , C be 2n disjoint simple closed curves bounding a commonregion D. Suppose that for each k, there is an element gk e loll, with gk(Ck) = Ck,and gk(D)f1D = 0. Use C.2 to prove inductively that G = <g,,...,ga> is dis-crete, free on these n generators, purely loxodromic, geometrically finite, and thatD U C, U U C. is a fundamental set for G (G is a Schottky group of rank n; G isclassical if all the C. are circles; (see X.H)). (Hint: for the induction step, drawa simple closed curve W in the common exterior of all the curves, where Wseparates C,, ..., Cq_, from C. and

C ..., C be as in F.8, except that for some k, we allow Ck and Ck tointersect at one point xk, and we require that the corresponding gk be parabolic(necessarily with fixed point at xk). Prove that the corresponding group G =

VII.F. Exercises 169

<g1,..., g") is discrete, free on these n generators, and geometrically finite; alsoprove that D U (C, - {x, } U U (C" - {x" }) is a constrained fundamental set forG, (if Cm fl c,, = o, then (x.) = 0) and that every parabolic element of G isconjugate to one of the parabolic generators (G is called a Schottky type group).

F.10. Some problem as F.8, but use E.5 for the induction step. (Hint: LetJ, =J2 = {1}, and let W, =C", W2 = C;).

F.1 1. Same problem as F.9, but use E.5 for the induction step.

F.12. Let Go = <z - z + 1), and let J, = J2 = Go. Let r be a complex number withpositive imaginary part. Let W, = {Im(z) = 0}, and let W. = {Im(z) = Im(r)}.Let f(z) = z + r. Use E.5 to show that G = <Go, f) = Go*f is discrete, freeAbelian of rank two, and that the usual fundamental parallelogram is a fun-damental domain for G.

F.13. Let Go = <z - e2"'"z), let J, = JZ = Go, and let f(z) = iz, Al l> 1. Use E.5with W , = { l z l = 1 } and W 2 = { I z l = I ).I } to show that G = <Go, f) is Kleinian,G = Go: f = 71 + Z. (the last equality should be read both as a group iso-morphism and as an equality of Kleinian groups (see V.F.4)), and that D ={zl 1 < Izl < 1;.1, 0 < arg(z) < 2n/n} is a fundamental domain for G.

F.M. Letj(z) = e2x'""z,g,(z) = l/z, g2(z) = k2/z, lkl > 1. Let G1 = <j,g1>, and letG2 = < j, g2) be two dihedral groups with the common subgroup < j ). Use C.2with W =I JzJ = 1k1112} to show that G = <G,,G2) is Kleinian, G = G, *J G2,and that G is geometrically finite. Also construct a fundamental domain for G,and show that G is a double dihedral group (see V.F.6).

F.15. Let G, and G2 be non-elementary Fuchsian groups acting on H2. Assumethat the positive imaginary axis L is a boundary axis for both G, and G2, whereG, has {Re(z) > 0} as its boundary half-plane, and G2 has {Re(z) < 0} as itsboundary half-plane. Assume also that J = StabG,(L) = StabG=(L) is hyperboliccyclic. Use the full circle containing L in C.2 to show that G = <G1, G2 > isFuchsian, and G = G, *J G2. Show also that H2/G is the union of lie/G1, withits boundary half-plane deleted, and H2/G2, with its boundary half-plane deleted,where these two surfaces are glued together along L/J.

F.16. Let Go be a non-elementary Fuchsian group acting on H2, and let J, andJ2 be hyperbolic cyclic subgroups. Suppose Go has boundary half-spaces H1 andH2, where H. is precisely invariant under J., but H1 and H2 are not Go-conjugate.Suppose further that J, and J2 are conjugate in M (that is; the geodesic lengthsof OH, 1J, and 3H2/J2 are equal). Let f be some element of PSL(2,18) wheref(8H1) = 8H2, and f(HI) fl H2 = 0. Show that G = <Go, f > is discrete andFuchsian, and G = Gosf. Also show that H2/G is H2 /Go with Hl/J, and H2/J2deleted, and the two boundaries sewn together.


F.17. Let F be a non-elementary Fuchsian group acting on H 2, where the positiveimaginary axis L is a boundary axis, and let h(z) = - z. Prove that G = <F, h>is Kleinian, and construct a fundamental domain for G, starting with a funda-mental domain for F. (Hint: Let J = Stab6(L), let GI = F, let G2 = <J, h>, andlet W in C.2 be the circle containing L). (Observe that [G2 : J] = 2.)

F.18. Let F be a non-elementary Fuchsian group acting on H2, where the positiveimaginary axis L is a boundary axis. Let J = Stab(L) be generated by z -+ A2z,d > 0. Use C.2 to prove that G = <F, z -. - Az> is Kleinian. Also construct afundamental domain for G, and describe Q(G)/G.

F.19. Let Go be the (2, 2, oo)-triangle group, and let J, and J2 be non-conjugatesubgroups of order 2. Use E.5 to construct a Kleinian group G Go so that allthe elliptic fixed points of G in 1-03 project to just one connected curve in theassociated 3-orbifold M. Can this curve be closed?

VII.G. Notes

The first combination theorem was due to Klein [40], and was utilized foruniformization by Koebe [41]; Klein's theorem, the free product, appears hereas A.13. More general combination theorems for Fuchsian groups, involvingamalgamated free products and HNN-extensions with cyclic distinguished sub-groups, were developed by Fenchel and Nielsen [24]. Versions for Kleiniangroups, also involving only cyclic distinguished subgroups, appear in [56] and[60]; these were generalized in [64]. The use of combination theorems forThurston's uniformization theorem can be found in [74].

Greater detail in the development of the amalgamated free product and theHNN-extension, from a purely group theoretical point of view, can be foundin [50].

B.19. General position arguments in 3-dimensional topology can be foundin [33].

Chapter VIII. A Trip to the Zoo

This chapter contains a collection of more or less specific Kleinian groups withdiverse properties. The organization is strictly utilitarian; the examples aregrouped according to the techniques used in their construction, rather than byrelated properties. Some of the examples are included for practical instructionin the use of combination theorems, or of Poincare's polyhedron theorem; othersare included for theoretical purposes, and will be used in subsequent chapters;yet others are included for the usual aesthetic reasons; the author finds themimportant, interesting, or amusing (however, there is no guide as to the reasonor reasons any particular exhibit is in the collection).

Some of the constructions yield families of Kleinian groups. We simplymention this as a fact, but do not pursue it. The study of families of Kleiniangroups requires the introduction of quasiconformal mappings and Teichmiillerspaces; an undertaking beyond the scope of this book.

VIII.A. The Circle Packing Trick

A.I. Let C,, C, ..., C,,, C be disjoint simple closed curves in the extendedcomplex plane bounding a common region D. Assume that for each pair of circles,{Cm, C,}, there is an element fm in ADO, so that fm(Cm) = and fm(D) n D = 0.If oo e D, then we can restate this last condition as being that f, maps the outsideof Cm onto the inside of We can use either combination theorem to showthat G = <f1,..., f,> is Kleinian, free, and purely loxodromic, and that D is afundamental domain for G (see IV.J.20, V.I.12, and VII.F.8, 10).

A.2. Now consider infinitely many pairs of disjoint circles C1, C;, ... as above.That is, the circles are pairwise disjoint, and no circle in the set separates 00 fromany other circle in the set. Choose elements fm in M, where f. maps the outsideof C. onto the inside of C,,. Let G = < f,, ... >. If G is discrete and if the circlesall have a common exterior D, where D is a fundamental domain for G, then wecall G an infinite Schottky group.

In the case that the circles do not all have a common exterior, that is, theclosure of the union of the insides of all the circles is all of C, then we still call Gan infinite Schottky group, provided it is discrete.

172 VIII. A Trip to the Zoo

Let g be some element of G. Only finitely many of the f,,, appear when weexpress g as a product of the generators f,, .... Hence we can use the resultsabout classical Schottky groups of finite rank (VII. F.8, 10) to conclude that (i) Gis free on the generators ft,..., (ii) G is purely loxodromic, and (iii) D is preciselyinvariant under the identity in G.

D need not be a fundamental domain for G (see H.2); we need conditions thatwill guarantee that it is; we give two different sets of sufficient conditions for thisto be so.

A.3. Proposition. If the circles Cm, C, all lie in a bounded part of C, and if for eachm, C. is the isometric circle off., and C,, is the isometric circle off.-', then D is afundamental domain for G.

Proof. Since the circles are all bounded, G is Kleinian, and D is precisely invariantunder the identity in G. Let D' be the Ford fundamental domain for G; It isimmediate that D' c D. Since D is precisely invariant under the identity in G,and every point of D is G-equivalent to some point of D', D = Y. 0

A.4. We need the following notation for the next proposition. For any pair ofdisjoint circles (C, e) let p be the (Euclidean) radius of the smaller circle, and let8 be the distance from the center of the smaller circle to (the boundary of) theother.

Proposition. D is a fundamental domain for G if the following condition holds. Thereis a number a > 0, so that for any pair of circles (C, C) from the set {C Ci,...b/p> I +a.

Proof. For each circle C, where C is either a C. or a C call the component ofC - C containing D the outside of C. Let S be the hyperplane in H3 spanning C,and let H be the half-space bounded by S spanning the outside of C. Let E bethe intersection of all these half-spaces; it suffices to show that E is a fundamentalpolyhedron for G.

Since E has no edges, the only condition of IV.H.I I that needs to be checkedis the completeness condition, IV.H.10, and for that, it suffices to show that if Sand S are any two sides of E, then the hyperbolic distance d(S, S) is bounded frombelow.

Let C and C be two of the circles from the set (C,, C; , ... }, let S and . be thecorresponding hyperbolic halfplanes; let C be the smaller of the two circles, andlet a be its center. It is clear that everything under consideration is invariantunder conjugation by Euclidean motions; hence we can assume that a is real andpositive, that the center of C is real and negative, and that the closest point onC to a is at the origin; i.e., a = S. Now consider only that part of C and Z` in theupper half-plane, and regard C and G` as being hyperbolic lines in H2. It is clearthat d(S, 3`) = d(C, C), where the first distance is in H3, while the second is in H2.

VIII.A. The Circle Packing Trick 173

a=b

Fig. VIII.A.1

Let A be the positive imaginary axis. It is clear that d(C, C) > d(C, A); henceit suffices to show that d(C, A) is bounded from below.

Let L be the (Euclidean) ray through the origin where L lies in the upper halfplane and is tangent to C. Since every dilation with fixed points at 0 and oo keepsboth A and L invariant, L, together with its reflection in the imaginary axis, isthe set of points at fixed hyperbolic distance from A. Since L is tangent to C,d(A, L) = d(A, C).

Write L = {z I arg(z) = 0}, and observe that sin(O) = p/b (see Fig. VIII.A.1).Let f be the radius of the circle centered at the origin and orthogonal to both Cand L; i.e., fi = (62 - p2)"2. On the circle of radius fi, write z = fleim, and compute

nn 1 + cos(0) b +d(A, L) = csc((p) d(p = log

sin(0) = logP

> log6

>- log(l + a).P

We have shown that any two sides of E are distance at least log(l + a) apart;hence E is a fundamental polyhedron. Then by VI.A.3, D is a fundamentaldomain. 0

We remark that the proof above shows that G is discrete, and that E is afundamental polyhedron for G, even in the case that D, the common exterior ofall the circles, is empty.

AS. Corollary. Every point of accumulation of the defining circles on aD is a limitpoint of G.

A.6. Proposition. Let T be an arbitrary domain in the plane. Then there is a sequenceof disjoint circles {C1, Ci,... }, where no circle in the sequence separates any otherfrom oo, each circle lies in T, the set of circles satisfies the hypothesis of A.4, andthe intersection of T with the common exterior of all the circles is empty.


Proof. We regard T as being a domain on §2, and use the spherical metric forall distances. Instead of labelling the circles as C,, C., ..., we label them as C1,C2, .... There may be more than one largest circle we can draw in T; pick one.Let its center be a and its radius p,. Let C, be the circle with center a, andradius p,/2.

Assume we have found the circles C,, ... , C. Let T. be the region T withthese n circles, and their interiors, deleted. Pick a largest circle that can be drawnin T,,; let its center be and its radius Let be the circle with center

and radius /2. It is clear that is not larger than any of the precedingcircles, so the ratio of distance S, from the center of C+, to any C, to the radiusp of is greater than 2.

Since the circles are disjoint and the sum of their areas is bounded, the radiusof the n-th circle tends to zero. Thus if z e T, and a > 0, the circle of radius a aboutz cannot be disjoint from all the C. 0

Let {C,, C;,... } be a set of circles as above, let fm map the outside of C. ontothe inside of C,,, and let G = <11,12,...> be the corresponding infinite Schottkygroup. Under these circumstances, we say that G fills up T Note that since T isopen, every point of 8T is a point of accumulation of the defining circles.

A.7. An arbitrary limit set

Proposition. Let X be any closed nowhere dense subset of C. Then there is aKleinian group G so that A(G) X.

Proof. Let R be the complement of X, let S be a non-empty open subset of R,where S is bounded away from X, and let T= R - S. Let G be an infiniteSchottky group which fills up T Since S :A 0, G is Kleinian. It is clear that everypoint of X is a point of accumulation of the defining circles for G, so by A.5,XaA. 0

A.8. Filling a regionLet G, be a Kleinian group, normalized so that oo a A, and let D, be a

fundamental domain for G,, where D, is the boundary of a fundamental polyhe-dron E,. Let T be a bounded open subset of D, . We fill T with an infinite Schottkygroup G2, defined by the set of circles {C,, C;,... }, where the inequality in A.4holds not only between any two defining circles for G2, but also holds betweenany defining circle for G2 and any side of D, (since E, is a fundamental polyhe-dron, the sides of D, are circular); it is easy to adjust the argument in A.6 toaccommodate this case.

Note that G2 is Kleinian, and that the complement of T is a fundamentaldomain for G2. Let E be the polyhedron formed by E, and the hyperbolic planesspanning the C. and C,. Since all the new sides of E are at finite hyperbolicdistance from each other, and from any side of E,, E satisfies the hypotheses of

VIII.B. Simultaneous Uniformization 175

Poincare's theorem. The group G, generated by the side pairing transformationsof E, clearly splits as a free product, G = G, * G,. Also aE = D, - T. We have con-structed a Kleinian group G, so that there is a conformal embedding of Q(G)/Ginto Q(G,)/G,, where the image of Q(G)/G is the image, under the natural pro-jection, of D, - T. If D, - T :A 0, then G is Kleinian. In any case, G is discrete.

A.9. Killing a componentLet G, be a Kleinian group where Q(G,)/G, has at least two components. Let

So be a component ofQ(G,)/G,, let D, be a fundamental domain for G,, whereD, is the boundary of a fundamental polyhedron for G and let T be the preimageof So in D,. Let G be the group obtained by filling T Then Q(G)/G is Q(G,)/G,with the component So deleted.

If we start with a finitely generated Fuchsian group of the first kind, and killone of the two components, then the new group G is analytically finite, but notfinitely generated (the infinite Schottky group is of course infinitely generated,and G is the free product of the Fuchsian group and the infinite Schottky group).This shows that the converse to Ahlfors' finiteness theorem is false.

A.1O. Cutting an edgeLet G, be a Fuchsian group of the second kind acting on 0-02. Let f be the

lower half-plane, and let S be an arbitrary sub-domain of 0_/G,. We construct aKleinian group G, where Q(G)/G = (0-02/G,) U S as follows. Choose a fundamen-tal polyhedron E, for G, , and let D, be the fundamental domain on the boundaryof E,. Choose Tc D, so that p(T) is the complement of S in L/G,. When we fillT, we obtain a new Kleinian group G, so that Stab(H2) = G,, and Q/G is thedisjoint union of 0-02/G, and S.

If we let T = D, n L then we obtain a group G with Q/G = 0-02/G,.We formally state the last remark.

Proposition. Let F be a Fuchsian group acting on H2. Then there is a Kleiniangroup G containing F, so that 0-02 is precisely invariant under F in G, and Q(G)/G =0-02/F.

Similarly, if G, is any Kleinian group, and So is a subsurface of Q(G, )/G,,where every connected component of the complement of So has non-emptyinterior, then we can fill that part of a fundamental domain lying over thecomplement of So so as to obtain a new Kleinian group G, containing Go as asubgroup, where Q(G)/G = So.

VIII.B. Simultaneous Uniformization

B.I. Let S be a marked Riemann surface, let p: U - S be its branched universalcovering, and let q: A -+ S be a branched regular covering given by a Kleinian


group, where both coverings are branched over the same points to the sameorders. Let G be the group of deck transformations acting on U; note that thereis a freely acting subgroup H c G, so that U/H = d (see III.F.8).

The sphere covers only itself, and, as an unbranched universal covering, theplane covers only itself, the punctured plane, and the tori; of these, only the planeand punctured plane are planar. We have shown the following.

Proposition. Let G be a non-elementary Kleinian group. Then every component of1?/G is, as a marked Riemann surface, hyperbolic.

B.2. We showed in A.10 that for every marked Riemann surface S, there is aKleinian group G so that Q(G)/G = S. We also noted above that if S is nothyperbolic, then there is no Kleinian group G so that Q(G)/G contains S andsome other surface.

B.3. Proposition. Let {Sm} be a sequence of hyperbolic marked Riemann surfaces.Then there is a Kleinian group G with 12(G)/G = U S..

Proof. For each m, let G. be a Fuchsian group, acting on B2, so that B2/Cm = Sm.For each Gm, choose a fundamental domain f),,, where {IzI > 5,,} U {oo} a b..Let

m-tAm=bm+2

k.l

and let jm(z) = z + L. Set G. = j_Gmj;', and set D. = jm(f)m). Then Dm, whichis just D. translated to the right by Am, is a fundamental domain for G. (see Fig.VIII.B.1, where m = 3, S1 has signature (1, 1; oo), S2 has signature (2, 0), and S3 isan annulus; the identifications of the sides are not shown in the figure).

The boundaries of the D. are all disjoint; to see this, let xm be the "center" ofDm, that is, xm = j.(0), let W. be the circle centered at xm of radius bm, and let B.be the inside of Wm. For fixed n, then circles W1, ..., W. form a chain of tangentcircles, where the outside of W. lies in Dm, and the inside of W. contains thecomplement of Dm. We conclude that for every n

Fig. VIILB.I

VIII.C. Elliptic Cyclic Constructions 177

we also see that

n

U Dm=C;M=1

n

,5. = n D.m=1

certainly includes all the sets D. fl jm(B2), m = 1, ... , n.Let n = <G,,..., Gn>. Our next goal is to inductively show that c is Klein-

ian; C. _ G,,; that jm(B2) is precisely invariant under G. in C,; and thatD = D, fl - - fl D is a fundamental domain for Cn. We have defined the circle W.so that the closed outside disc bounded by W. is precisely invariant under theidentity in G. By the induction hypotheses, the closed inside disc is preciselyinvariant under the identity in C.-,. In fact, both of the last two statementsremain true if we replace the closed disc by a slightly larger open disc; it followsthat each of the closed discs is a strong { I }-block. The desired result now followsfrom VII.C.2.

Set G' = <G1, G2, ... > = <01,... t;,,, ... >, and set D = n D,,. Since every ele-ment of G' lies in some Cr., D is precisely invariant under the identity in G', andfor every m, jm(B2) is precisely invariant under G. in G'. However, this does notguarantee that D is a fundamental domain for G'.

Let E be a fundamental polyhedron for G', and let °aE be the boundaryfundamental domain. Let S be the set of translates of U jm(B2), and let T be theinterior of its complement in 1E.

Let G be the group obtained from G' by filling T Exactly as in the proof ofA.6, we can fill T so as to guarantee that °aE - T is a fundamental domain forG. Hence Q(G)/G is the disjoint union:

Q(G)/G = Ujm(B2)IGm = U Sm. 0

VIII.C. Elliptic Cyclic Constructions

C.I. The double dihedral groupsLet G, be a v-dihedral group, normalized so that the element of order v has

its fixed points at 0 and oo, and so that g, (z) = 1 /z lies in G, . Then all thehalf-turns in G, have their fixed points on the unit circle. Choose a constrainedfundamental set D, for G, so that the interior of D, is the "half-sector" {z 10 <arg(z) < 2n/v,IzI > 1}.

Let G2 also be a v-dihedral group, normalized so that the element of order vhas its fixed points at 0 and oo, but normalized so that g2(z) = 9/z lies in G2.Then all the half-turns in G2 have their fixed points on the circle of radius threeabout the origin. Choose a constrained fundamental set D2 for G2 so that theinterior of D2 is the "half-sector" {zl 0 < arg(z) < 2a/v, IzI < 3} (see Fig. VIII.C.1).


Fig VIII.C.I

Let J be the common subgroup of G, and G2 generated by j(z) = 2,d/VZ.

Choose E = {z 10 S arg(z) < 2n/v} as a constrained fundamental set for J. Ob-serve that B, = {z) IzI > 2} U {oo} is precisely invariant under J in G,, whileB2 = {zI IzI S 2} is precisely invariant under J in G2. In fact, somewhat largerdiscs are precisely invariant, from which it follows that B. is a strong (J, Gm}block. By VII.C.2, the group G = <G,, G2> is Kleinian, G = G, * G2, and D =D, fl D. = (D, fl B2) U (D2 fl B,) is a constrained fundamental set for G.

Note that d = {zI I < IzI < 3,0 < arg(z) < 2a/v}. There are four non-conjugate elliptic fixed points on 8D; for example, at 1, ei"'", 3ei" J" and 3. Foldingtogether the sides of D, we see that Q(G)/G has signature (0,4; 2, 2, 2, 2). Alsoobserve that the circle {IzI = 2) projects to the v-th power of a simple loop thatseparates the projections of I and eixi" from those of 3 and 3ei*"".

Since every element of G stabilizes the pair of points {0, co), G is elementary;in fact, G is a double dihedral group.

C.2. There are many non-conjugate double dihedral groups, even for the samev. In order to construct them, we make use of the normalizer of J in M.

First there are the rotations h,(z) = ei`z, t real. Almost everything in theconstruction above remains unchanged if we keep G, fixed and replace G2 byh, G2 h-' . The only thing that is changed is the constrained fundamental set D2;the fundamental domain,132, is unchanged as a set; the identifications of the sidesare changed, and the elliptic fixed points occur at different places (see fig. V.F.2).This operation is called sliding.

J also commutes with every hyperbolic transformation k,(z) = tz, t > 0.Keeping G, unchanged, and replacing G2 by k,G2k, ' may however affect the useof the combination theorem. We no longer have {IzI = 3} as the circle dividingC into two discs, but we now must use { IzI = 3t}. For t > 1/3, this does not causeany significant difference. For t = 1/3, the construction breaks down, and thecombined group is either not discrete, or is not the amalgamated free product(this is similar to the construction in VII.A.8). Fort < 1/3, the construction againworks, with the roles of G, and G2 interchanged. This operation is called bending.

Notice that we have used up our normalization possibilities for G,. Startingwith fixed G, and G2, the different groups obtained by sliding or bending, orboth, are indeed different, provided we keep our bending parameter t > 1/3. Thatis, in general, one is a deformation of the other, but they are not conjugate in M.


D,

Fig. VIII.C.2

In fact, every double dihedral group can be obtained from the original one, upto conjugation in M, by a combination of both bending and sliding.

C3. We next illustrate the use of the first combination theorem in a more generalsetting. Let G, be a finitely generated Fuchsian group of the first kind containingan elliptic element of order v. Normalize G, so that it acts on B2, and so thatStab(O) = Stab(oo) = J has order v. Then there is some number S > 0, so thatthe disc B, = { I z 15 61 is precisely invariant under J in G,. Let D, be a con-strained fundamental set for G,, where D, is maximal for B,, and D, is containedin the "natural" constrained fundamental set for J: E = (z 10 5 argz < v} (seeFig. VIII.C.2 for the case that G, has signature (1, 1;4), and OB1 is labelled "W").

Next let G2 be some group just like G,; that is, G. is a finitely generatedFuchsian group of the first kind containing a maximal cyclic subgroup of orderv. First normalize G2 so that it acts on B2, and so that J is a maximal cyclicsubgroup of G2. Then there is a b' > 0, so that I z I 5 b' is precisely invariantunder J in G2. Of course, it is equally true that Izl z 1/6' is precisely invariantunder J in G2. Now conjugate by a dilation of the form z -+ tz, 0 < t < 1, so thatG2 operates on a smaller disc, also centered at the origin, and so that B2 ={z J I z I z 6) U { oo } is precisely invariant under J in G2. Choose a constrainedfundamental set D2 for G2 which is maximal for B2, and which is contained in E.(see Fig. VIII.C.3 for the case that G2 has signature (0, 3; 4, oo, oo)).

With G, and G2 normalized as above, J is a common subgroup, and B. is astrong (J, G.)-block. Hence VII.C.2 is applicable. Set G = <G1, G2>, and setD = (L) , fl B2) U (D2 fl B,), observe that D is constrained (see Fig VIII.C.4 for theamalgamation of the groups shown in Figures VIII.C.2-3). It is easy to see thatQ(G)/G has three components; one of these is the projection of the outside of theunit disc (this is identical with the projection of the outside of the unit disc under


Fig. VIII.C.3

Fig. VIII.C.4

G,), a second is the projection of the inside of the circle of radius t (this is identicalwith the projection of the inside of the circle of radius t under G2), and the thirdcan be obtained from the other two components of Q(G,)/G, and 12(G2)/G2 bycutting out a disc containing the special point of order v from each of them,and sewing together the boundaries of these two discs. For the groups shown inFigs. VIII.C.2-4, these three surfaces have signatures (1, 1; 4),(0,3; 4, oo, 00), and(1,2;oo,oo)

Observe also that the inverse image of the third component listed above isG-invariant. The circle W of radius b lies in this component and projects to asimple loop which, when raised to the v-th power, lifts to a loop.

CA. For the next construction, let Go be a Fuchsian group, acting on 132, where132/G0 has exactly one point of ramification of order v (in Fig. VIII.C.2, v = 4,


Fig. VIILC.5

and Go has signature (1, 1; 4)). Normalize so that j(z) = e2i a Go. For b suffi-ciently small, the set B, = (z I IzI 5 b) is precisely invariant under J in Go. Sinceevery element of Go commutes with reflection in S', it is equally true that B2 =(zI IzI z 1/8) U (ao) is precisely invariant under J in Go. Choose a constrainedfundamental set Do for Go that is maximal with respect to B, and B2; that is,Do fl B_ is a fundamental set for the action of J on B,,,.

Let f(z) = 52z. Since (B,, B2) is precisely invariant under (J, J) in Go, and sincef commutes with every element of J, B, and B2 are jointly f-blocked.

Let G = <Go, f>, let A be the open annulus between B, and B2, and letD = (Do fl A) U (Do fl aB1). Then by VII.E.5, G = Gos f, G is geometrically finite,every non-loxodromic element of G is conjugate to an element of Go, and D is afundamental set for G (see Fig VIII.C.5). Note that with the choice of Do as givenin Figure VIll.C.2, D is constrained, and we can read off from D that Q(G)/G isconnected, and consists of Q(G0)/Go, where we have cut out discs about the twospecial points of order v, and glued together the boundaries of these discs. ForGo as in Fig VIII.C.2, Q(G)/G has signature (2, 0) (see Fig VIILC.6).

Note that the fixed points of J are also the fixed points off hence these pointsare limit points of G. If Go is non-elementary, then so is G. We have constructeda non-elementary Kleinian group containing elements of finite order, where theprojection from D is unramified.

Notice also that all the <f>-translates of the unit circle are limit points of G.Hence 0 and oo, which are loxodromic fixed points of G, do not lie on theboundary of any component of G.

CS. There are also versions of sliding and bending for the above operations, evenin this simple case, where J, = J2. That is, we slide by replacing f(z) = A2z byf (z) = ea12z; we bend by replacing f(z) = A2z by f,(z) = tA2z, t > 0.


Q(Go)/Go

S2(G)/G

Fig. VIII.C.6

Fig. VIILC.7

C.6. In order to illustrate the use of the second combination theorem in a moregeneral setting, we start with a particular double dihedral group G0, where Go isgenerated by j(z) = -z, g(z) = 16/z, and h(z) = 400/z. Choose a constrainedfundamental set Do for Go, where int(D0) = {zJ4 < (zI < 20, -n/2 < argz <a/2} (see Fig. VIII.C.7). Choose disjoint closed circular discs B1, containing thepoint 4, and B2, containing the point 20, so that (B1, B2) is precisely invariantunder (J1,J2) = (<g>, <h>) in Go; for example, let Bt = {z) Iz - 51 5 3), and letB2 = {z I Iz - 251:9 15}. Let fl be a transformation mapping the outside of B1onto the inside of B2; for example, let fl = (25z - 80)/(z - 5) (note that B2 canbe obtained from Bt by applying k(z) = 5z; fl is the composition of reflection in8B1, followed by complex conjugation, followed by k). It is easy to see that thestrong blocks B1 and B2 are jointly f1-blocked


Fig. VIII.C.8

Fig. VIII.C.9

Let G1 = <Go,f1 >. The fundamental domain DI for G1, given by VILE.5 isshown in Figure VIII.C.8. D1 is contained in the sector {z I - n/2 < arg(z) < n/2}.We cut along the dotted line, and paste using j to obtain a fundamental domainD'1 contained in the sector {z 10 < arg(z) < n}; see Figure VIII.C.9.

Let Bp, be obtained from B. by rotation about the origin through an angleof n/2. Let f2(z) = (25iz + 80)/(z - 5i), so that f2 maps the outside of 01 onto theinside of B2. Since B. fl B'. = 0, one easily sees that (91, B2) is precisely invariantunder (g o j, h o j) in G1, and that 91 and 92 are jointly f2-blocked. The fun-damental domain D, given by VII.E.5, for G = <Gl,f2> is shown in FigureVIII.C.10. D has 12 sides, and the angle between every pair of adjacent sides isn/2. It is easy to check that every point of 8D lies in °Q(G) hence *D(G) = 92(G).


Fig. VIII.C.10

Fig. VI I.C.I I

Fold together the sides of D to see that DIG is a closed Riemann surface ofgenus 2.

We have constructed a geometrically finite non-elementary Kleinian groupG with torsion, where °SQ(G) = Q(G); that is, every fixed point of an ellipticelement of G is also a limit point of G, and Q(G)/G is both compact and connected.

One can rind a presentation for G in two different ways. First, by usingVII.E.5(i); second by constructing spanning discs in H' for the sides of D,and using Poincare's theorem. In any case, set a 1 = f b, = h, a2 = h O j andb2 = f2, to obtain the presentation: <a b, , a2, b2 : n b; = a2 =(b1 oa2)2 = 1>

C.7. Let G be the group constructed above, and let H be the subgroup generatedby g o h, fl, j -fl -j-1, f2, and j o f2 o j-t . We easily observe that H is a classicalSchottky group (see Figure VIII.C.11 for the fundamental domain giving this)of rank 5; in particular, Q(H)/H is a closed Riemann surface of genus 5. We alsonote that H is of finite index in G.

VIII.D. Fuchsian Groups of the Second Kind 185

VIILD. Fuchsian Groups of the Second Kind

D.1. Let C1, Ci, ..., C., C, be 2n disjoint circles where each C,,, and each C;,, isorthogonal to the unit circle 5', and every one of these circles lies outside all theothers. For each m, let f,,, be an element of M, where f_(C.) = C,, f,, maps theoutside of C. onto the inside of C,,, and f,, preserves the unit disc 132; i.e., f,, a P2+.

For example, one could choose f,, to be reflection in C. followed by reflectionin the (hyperbolic) perpendicular bisector of the common perpendicular to C.and C,.

As we saw in A.1, one can use either combination theorem to conclude thatG = <ft,..., f,> is Kleinian, free on these n generators, purely loxodromic,geometrically finite, and that D', the region outside all of these circles, is afundamental domain for G.

Take a closed orientable surface of genus n, and smoothly embed it in S3. Itdivides S3 into two homeomorphic compact 3-manifolds with boundary; these3-manifolds are called handlebodies of genus n. It is easy to see that the 3-manifoldM = Iii/G is homeomorphic to a handlebody of genus n. This result is true forall Schottky groups on n generators; it is independent of the size and relativepositions of the 2n circles (or simple closed curves, if the group is not classical).

Since D' is invariant under reflection in S', D = D' f1132 is a fundamentalpolygon for the action of G on 132. There are many different possible arrange-ments for the identifications of the sides of D. In any arrangement, it is easy tosee how many boundary components S = B2/G has, and then compute the genusof S from the knowledge that Q(G)/G has genus n. If the genus of S is p, and thenumber of boundary components is k, then 2p + k - I = n (see Fig. VIII.D.1 forthe case that p = 1, k = 1, and see Fig. VIII.D.2 for the case that p = 1, k = 2).

Fig. VIII.D.1

Fig. VIII.D.2


D.2. In the construction above, there is a direct relationship among the hyper-bolic distances between certain of the circles, and the multipliers or traces of theprimitive boundary elements. Assume that the multipliers lie between 1 and oo.Then, as the circles get smaller and farther apart, the multipliers get larger; also,keeping the Euclidean size of the circles fixed, the multiplier approaches 1 as thecircles get closer together. Using this information, it is not hard to show theexistence of a Fuchsian group G of the second kind, where DIG has genus p,with k boundary components, and the multipliers of the k primitive boundaryelements are arbitrarily given between 1 and oo, provided, of course, thatn = 2p + k - I > I (see H.7- 10).

Renormalize so that G acts on 0-92, and so that the second quadrant is aboundary half-space for G; let J be the stabilizer of this boundary half-space.Then there is a minimal number b > 0, called the angle width of J in G, so thatthe sector ( zI6 < arg(z) < n) is precisely invariant under J = Stab(0) = Stab(oo).Of course, 0 < h 5 n/2.

Proposition. Let G be a Fuchsian classical Schottky group, as above. Let J be aboundary subgroup of G, and let S, 0 < S < n/2, be given. Then there is a Fuchsianclassical Schottky group G', where B2/G' is topologically equivalent to 032/G, andthe angle width of the corresponding subgroup J' in G' is less than 6.

Proof. Normalize so that the boundary axis is the positive imaginary axis L, andso that none of the defining circles lie entirely in the left half-plane. Then thereare exactly two circles, call them C and C', intersecting L. Find a new classicalSchottky group (;, with defining circles C,,, C,,, where these have the sameidentifications, as the corresponding C. and C,, as follows. Let C and C' be thecircles corresponding to C and C'. Construct these circles so that C and C' arethe only circles which intersect L, and so that the (Euclidean) distance betweenC and C' along the positive reals is small compared to their diameters. Then theray {arg(z) = S; intersects only C and C' of the defining circles. It easily followsthat the set {z 16 5 arg(z) 5 n} is precisely invariant under Stabd(L). p

D.3. There is a construction analogous to that of D.1, using parabolic and ellipticgenerators. A parabolic pair consists of two hyperbolic lines in H2 that are tangentat oo; that is, as a pair of Euclidean circles, the parabolic pair consists of twotangent circles, each orthogonal to the real axis. The point of tangency isnecessarily on the real axis. The two Euclidean circles are called the full parabolicpair.

A (full) parabolic pair divides 032(C) into three regions; the outside region hasthe full parabolic pair on its boundary. Normalize so that the point of tangencyis at oo, and so that the parabolic pair lies on the lines Re(z) = 0 and Re(z) = 1.Then j(z) = z + I maps one of these lines onto the other, and the outside regionD = j z 0 < Re(z) < I } is a fundamental domain for <j>.

VIII.D. Fuchsian Groups of the Second Kind 187

An a-elliptic pair consists of two hyperbolic half infinite line segments, startingat some point z e H 2, where they meet at the angle a, and each line ends at somepoint on the circle at infinity. The full a-elliptic pair consists of the a-elliptic pair,together with both its reflection in the real axis, and the two points on the circleat infinity; that is, the full a-elliptic pair consists of two of the four arcs of a pairof intersecting circles, the circles are both orthogonal to OH', and they meet atthe angle a. Necessarily, the two points of intersection are conjugate points withrespect to the real line.

An elliptic pair (full elliptic pair) divides H' (C) into two regions; in the outsideregion, the two lines meet at the angle a. In the other region, the inside region,they meet at the angle 2n - a. Normalize so that the two points of intersectionare at 0 and oo, and so that the full elliptic pair consists of the rays arg(z) = 0,and arg(z) = a. Then j(z) = e'°z maps one of these rays onto the other, and, ifa = 2n/q, q e Z, then the outside region D = {z 10 < arg(z) < a} is a fundamentaldomain for <j>.

D.4. We conclude this section with an explicit inductive procedure for construct-ing a Fuchsian group of the second kind with given signature. We start with atopologically finite marked Riemann surface S, where S has at least one boundarycomponent that is not a point, and if S has genus 0, then it has at least threeboundary components. Given such a surface S, we construct a Fuchsian groupF so that H2/F is topologically equivalent to S; more precisely, there is ahomeomorphism between H2/F and S that preserves the type of each boundarycomponent, and preserves the marking at those boundary components that arepoints.

Suppose S has genus p, n special points, with marking a,, ..., an, and m > 0holes. Start with a classical Schottky group Go, so that H2/Go is a surface ofgenus p, with m holes; then Go is a free group on n = 2p + m - I generators. LetDo be the fundamental domain defined as the outside of the 2n circles used todefine Go, and normalize Go so that the entire right half plane is contained in Do.If p = 0, and m = 1, then Go is trivial.

If a, < oo, choose L, as a full 2n/a,-elliptic pair entirely contained in theright half plane, where the outside D, of L, contains oo. If a, = oo, Let L, be afull parabolic pair, also completely contained in the right half plane, where theoutside D, of L, contains oo. Let g, a 12+ be an elliptic or parabolic transforma-tion identifying the sides of L1, and let G, = <g, >.

Find a circle W in Do fl D, that separates L, from all the defining circles ofGo. The two closed discs bounded by W are precisely invariant under the identityin Go, and G,, respectively. Hence G, = <Go, G, > = Go * G1, and all the otherconclusions of VII.C.2 also follow. In particular, if we let B. be the constrainedfundamental set obtained from D. by the addition of appropriate boundarypoints, then 61 = B, f' B2 is a constrained fundamental set for G,.

Note that we could also use Poincare's polygon theorem in H2 to realize thatG, is Kleinian, that into, fl H2) is a fundamental polygon for C1, and hence


that H2/V1 is a surface of genus p, with m holes, and with one special point oforder at,

If n > 1, choose L2 to be a full elliptic or parabolic pair, as above, where L2lies entirely to the right of L and the outside of L2 contains oo. Let G2 be anelliptic or parabolic transformation identifying the sides of L2, and let G2 = <92 >.Let D2 be the outside of L2 and let 132 be a constrained fundamental set obtainedfrom D2 by adjoining appropriate boundary points. Let W2 be a circle containedentirely in int(6, fl f32), where W2 separates the complement of 6, from thecomplement of 132. Then, as above, each of the discs bounded by W2 is preciselyinvariant under the identity in, respectively, G, and G2. Let 0, = <f;,, G2>, andlet 62 = 6, f162. Then by VI I.C.2, C2 = t;, * G2, 62 is a constrained fundamentalset for 02, and all the other conclusions of VII.C.2 are also valid.

As above, we can also use IV.H, to observe that int(62 fl lO2) is a fundamentalpolygon for 1;2, and that H2 /t;2 is a surface of genus p, with 2 holes, and twospecial points, one of order a, , and one of order a2.

We continue as above, until we reach C., and H2/0 is a surface of genus p,with m holes and n special points, of orders a,... I a,,.

VIII.E. Loxodromic Cyclic Constructions

E.I. Purely hyperbolic Fuchsian groups of the first kindLet G, be a purely hyperbolic Fuchsian group of the second kind, such as

one of those constructed in the previous section, where 0.12/G1 has genus p, withone boundary component. Assume that G, is normalized so that the secondquadrant is a boundary half-plane. Let J be the corresponding boundary sub-group; note that J = Stab({0, co}) is hyperbolic cyclic. Regarding G, as aKleinian group, the sector {n/2 5 arg(z) 5 3n/2} is precisely invariant under Jin G,. Then there is a somewhat larger precisely invariant sector: A, _{zI t/2-b, <arg(z)<3n/2+b,}, 61,>0.

Similarly, let G2 be a purely hyperbolic Fuchsian group of the second kind,where 0-02/G2 is a surface of genus p2 with one boundary component, and G2 isnormalized so that the first quadrant is a boundary half-plane. Assume that theboundary subgroup of G2, stabilizing 0 and oo, is the same group J as for G,.As above, there is a62 > 0, so that the sector A. = {zI -it/2- S2 < arg(z) <n/2+ S2 } is precisely invariant under J in G2.

Regard G, and G2 as Kleinian groups, and observe that the imaginary axisdivides C into two closed discs. The closed left half-plane, which we now call B1,is precisely invariant under J in G,, and the closed right half-plane, which wenow call B2, is precisely invariant under J in B2. The observations above aboutthe slightly larger precisely invariant sectors imply both that B. is a strong(J, G,.)-block, and that (b,,62) is a proper interactive pair.

Let G = <G,, G2>, and let j(z) = i.2z generate J, where A. > 1. Choose theconstrained fundamental set E = {z I )F' < IzI < ;.I for J, and choose con-

VI1I.E. Loxodromic Cyclic Constructions 189

Fig. V1II.E.1

Fig. VIII.E.2

strained fundamental sets D. for G. so that D. is maximal with respect toD. c E, and D. is invariant under reflection in the real axis; this last is easy toachieve, since reflection in the real axis commutes with every isometry of H2(pt = 1 in Fig. VIII.E.1, and p2 = 2 in Fig. VIII.E.2). With these choices, D.contains the intersection of A. with E. By VII.C.2(vii), D = Dt fl D2 = (D, fl B2) U(D2 fl B,) is a fundamental domain for G (see Fig. VIII.E.3).

Let S. = Q(Gp,)/G.. We can realize each of these surfaces as being the unionof three pieces. One of these pieces is an annulus; on S, it is the projection of thesector {z 1 n/2 < arg(z) < 3n/2}; the complement of the union of the translates


Fig. VIII.E.3

Fig. VIII.E.4

of this sector has two invariant connected components, the one in the upperhalf-plane projects onto a surface St. t of genus pt, with one boundary com-ponent, and the other component, in the lower half-plane, projects onto a surfaceSt. 2, also of genus p1, with one boundary component (note that the closure ofthe preimage of St. I in H2 is the convex region for GI). The surface St. I isconnected to the annulus along the projection of the positive imaginary axis, andthe surface St,2 is connected to the annulus along the projection of the negativeimaginary axis (see Fig. VIII.E.4, where pt = 1). Similarly, S2 is the union of anannulus (the image of the sector {zI ---x/2 < arg(z) < n/2}); a surface S2,1, ofgenus p2, with one boundary component (S2.1 is the image of the interior of theconvex region for G2 in B2); and 52.2, also of genus p2, with one boundarycomponent (S2, 2 is the image of the interior of the convex -egion for G2 in the

VI II.E. Loxodromic Cyclic Constructions 191

Fig. VIII.E.5

i S f Ot /;I=

Fig. VIII.E.6

lower half-plane). S21 , is joined to the annulus along the image of the positiveimaginary axis, and S2, 2 is joined to the annulus along the projection of thenegative imaginary axis (P2 = 2 in Fig. VIII.E.5).

To obtain 12(G)/G, we remove the two annuli, join St,, to S21,, and join St.2to S2221 to obtain two surfaces, each of genus pt + P2 (see Fig. VIII.E.6).

E.2 Sliding

We can slide the above construction by keeping Gt fixed, and conjugating G2by a hyperbolic transformation of the form f ,(z) = e'z, t real. This gives us a newgroup G2,, = f G2 f -', for which the same ball B2 is precisely invariant under thesame subgroup J = f Jf -'. Hence VII.C.2 is still applicable, but of course thefundamental sets D2 and D will change.

We can also regard this operation in terms of the relationship between thesurfaces H2/G and H2/G where G, = <Gt,G2.,). Let w be the image of thepositive imaginary axis on S = H2/G. As we saw above, w is a simple geodesic


that divides S into two pieces; call them S, and S2. Cut S along w, keep St fixed,and rotate S2 by a suitable amount (if one thinks of this rotation as being througha certain angle, so that rotation by 2n brings us back to where we started, thenthis angle of rotation is 27tt/a.), then glue the two surfaces back together to arriveat S, = l-O2/G,.

E3. A quasifuchsian group-bendingA quasifuchsian group is an analytically finite Kleinian group G, where Q(G)

has exactly two components, both invariant (the definition given in IX.B.2 is onlyapparently weaker, the equivalence of the two definitions is given in IX.F.10).We can construct a quasifuchsian group by bending the construction in E.1; thatis, we leave G, alone, and we conjugate G2 by a small rotation. Let h(z) = e"z,where b is real, and 151 is smaller than both b, and b2. Let Gz = hG2h-1, and letG' = <G1,GZ>. Notice that the same subgroup J is a common subgroup of G,and G2, that the same disc B2 is a strong (J, G2)-block, and that ($,, Lf2) is a properinteractive pair for (G,, Gz). Hence we can again apply VII.C.2. Let D2 = h(D2),so that D2' is a fundamental domain for G2, and D2 is also contained in E (seeFig. VIII.E.7). Let G' = <G,, G2 >; then G' is discrete, but no longer Fuchsian,G' = G, *, G2, and D'=D1 fl D2 is a fundamental domain for G' (see Fig.VIII.E.8). Comparing figures VIII.E.3 and VIII.E.8 we see that D(G')/G is ho-meomorphic to Q(G)/G.

The fundamental domain D' has two connected components; call them D,and D2. Each side of D, is paired with another side of D;,, and for both Di andD2, the identifications of the sides generates G'. We conclude that the connectedcomponent A. of Q containing D;, is G'-invariant. It follows that Q(G') hasexactly two components, both invariant; i.e., G' is quasifuchsian.

Fig. VIILE.7

VIII.E. Loxodromic Cyclic Constructions 193

Fig. VIII.E.8

E.4. Our next collection of examples starts with a Fuchsian group G, of thesecond kind acting on H2, where the second quadrant is a boundary half-space.Choose G, so that the sector {z 16 5 arg(z) < 2a - 6) is precisely invariant underJ = Stab({0, oo}) in G1, where 6 < 2n/n, for some positive integer n (see D.2).Let j(z) = t2z be a generator of J, where t is real and greater than 1. Choosethe fundamental domain E for J to be the annulus {zIt` < Izi < t}. Choosea fundamental domain D, for G, where {z 13 5 arg(z) S 2a - 6) c D, c E (inFig. VIII.E.1, n = 5).

Also choose an integer k, 0 < k < n, with (k, n) = 1. Define jo(z) = e2"a'"t2/"z.Let G2 = < jo>; observe that (jor = j, so that J is a subgroup of G2 of index n.Let D2 = {z I -n/n < arg(z) < n/n, t_, < Izi < t}.

It is almost immediate that D2 is a fundamental domain for G2; think ofC - {0} as being divided into sectors of the form {e2"`"''" < arg(z) <Each of the translates (jo)'(D2), m = 1, ..., n, lies in a distinct such sector.Looking inside any one sector, the J-translates of (jor"(D2) cover the sector.

It is an exercise to construct the sides of D2 so that they are pairwise identifiedby elements of G2; observe that there are necessarily exactly six sides (see Fig.VIII.E.9 for k = 3, n = 5).

The two rays (arg(z) = ±8} define a simple loop W in C, dividing C intotwo closed discs: Bt = {z 13 S arg(z) 5 2a - S}, and B2 = {zI -b S arg(z) Sd}. Note that b,,, is precisely invariant under J in although B2 is not preciselyinvariant under J in G2. However B. is a strong (J, G,,)-block, and ($,,1§2) is aproper interactive pair. In this case, D = D, fl D2 = (D, fl B2) U (D2 fl B2), thefundamental domain for G = <G1, G2> given by VII.C.2(vii), has two connectedcomponents, but there is a power of jo that identifies one side of the uppercomponent of D with a side of the lower component; hence S2/G has only one


Fig VIII.E.9

Fig. VIII.E.10

component (see Fig. VIII.E.10). We remark that for this group G, every com-ponent is simply connected, and no component is invariant.

The remarks below hold for any two coprime positive integers k and n,provided n > 1. For simplicity in bookkeeping, we now assume that k = 1.

There are some distinguished subgroups of G. Let d, be the component ofG containing the upper part of D, so that d, is contained in the sector {z 10 <arg(z) < 2n/n}. Sincej identifies two of the sides of this part of D, d, is J-invariant.Let G, be the stabilizer of d,. Looking at the identifications of the sides of D,one sees that G, = (G,,joGijo'> (a fundamental domain for the action of G,on d, is shown in Fig. VIII.E.11). The rays {arg(z) = 8} and {arg(z) = -b}together form a simple closed curve that separates C into two discs, one preciselyinvariant under G and the other precisely invariant under joG, jot. Exactly asin E.3, we can use these facts to show that G, is quasifuchsian.

For every m = 1, ... , n, let H. = joG, jo '. These are all distinct componentsubgroups of G, they all contain J, and they each stabilize a distinct component

Vll1.E. Loxodromic Cyclic Constructions 195

Fig. VIII.E.I I

of G. Hence j keeps at least n distinct components of G invariant. For each in,the intersection of H. and H,,, _1 contains many loxodromic elements. Theseloxodromic elements stabilize both components A,,, = jo (d 1) and A. _l , and theirfixed points lie on the common boundary ofd. and d,_I. It is easy to show thatthere are no other J-invariant components using the following construction. Foreach in, find a loxodromic transformation f, that stabilizes both A, and A,-,(modulo n). In A,, find an fm-invariant simple path, and an f,,,+I -invariant simplepath; connect these two paths by another path. The union of all these paths,together with the fixed points of the fm forms a set that separates 0 from co; henceevery J-invariant curve must cross this set. Since, except for the fixed points ofthe fm, this set lies entirely in A, U U d., there can be no other J-invariantcomponent of G. We have shown that G has exactly these n J-invariantcomponents.

If n = 2, then G has exactly two components. If n > 2, then by V.E.9, G hasinfinitely many components. In any case, the loxodromic element jo keeps nocomponent invariant, but j = (jo)" keeps n components invariant.

ES. An extended Fuchsian groupIf n = 2 in the construction above, then there is only one of these component

subgroups H = H1 = Note that H has two invariant compo-nents. Since G = <H, jo>, jo a H, and j0Hjo 1 = H, H is of index 2 in G. In thiscase, G is a Z2 extension of a Fuchsian group, but is not itself Fuchsian.

E.6. A purely loxodromic extended Fuchsian groupIn the example above, we constructed a Z2 extension of a Fuchsian group.

In this section, we give an alternate construction for such a group, and then inE.7, modify the construction so as to obtain an extended quasifuchsian group.We start with a Fuchsian classical Schottky group Go, acting on H 2, where H 2 /Go


Fig. VIII.E.I2

is a surface of genus p with two boundary components, and both boundarycomponents have the same geodesic length; i.e., every primitive boundary ele-ment of G has the same multiplier (all multipliers are assumed to be greater than1). Choose a fundamental domain Do for Go, where two of the side pairingtransformations, j, and j2, are non-conjugate primitive boundary elements (seeFig. VIII.E.12 for the case p = 1).

Let W. be the complete circle containing the axis of jm; W. bounds a closeddisc B. so that (B B2) is precisely invariant under (<A>, <j2>) Letf be some element of M mapping the outside of W, onto the inside of W2, andmapping the fixed points of j I onto those of j2.One easily sees that for any suchchoice of f, (BI, B2) is precisely invariant under (J,, J2) = (< j1 >, 02>), and thatthe Go-translates of B, U B2 do not cover all of Q(G0). Note that W, and W2 areboth orthogonal to t'; since f maps the fixed points of J,, which lie on t', tothe fixed points of J2, which also lie on t', f(t') = V.

Let G = <Go, f >. By VII.E.5, G is Kleinian, purely loxodromic, geometricallyfinite, and G = Go *f . Since t' is closed and G-invariant, it contains the limit setof G (see V.E. 4).

In fact every point of t' is a limit point of G. To see this, observe that sinceGo has only two non-conjugate boundary subgroups, Do - (B1 U B2) is boundedaway from t' . Then by VII.E.5(viii), no point oft' lies in °Q(G); since G is torsion-free, A (G) = 1.

Pick a point z on W1 n H 2 and observe that G is Fuchsian if and only iff(z) e H2. If f(z) # H2, then since G has exactly two components, f interchangesthese two components, and G has a subgroup of index 2, namely Stab(H2), thatis Fuchsian.

E.7. A purely loxodromic extended quasifuchsian groupThe construction above can be modified slightly so as to obtain an extended

quasifuchsian group. Let Go, J1, J2, Do, and W1 be exactly as above. Let W2 be


Wt

Fig. VIII.E.13

a circle close to W. and passing through the fixed points of J2 (see Fig. VIII.E. 13,where again p = 1). Let k be the closed inside of 1' Y2. If W2 is sufficiently closeto W2, then it remains true that (B1,22) is precisely invariant under (J1,J2).Choose some f e M mapping the inside of B1 onto the outside of O2, wheref maps the fixed points of J1 onto those of J2, and f maps some point ofH2 fl W1 to a point of W2 lying outside H2. We again note that the Go-translatesof B1 U B2 do not cover all of Q(G0) (i.e., there is a maximal constrained funda-mental set Do for Go that has non-empty intersection with the complement ofB1 U B2). Hence we can again apply VII.E.5 to conclude that a = <Go, f> isKleinian and purely loxodromic, ? = Go *1, and that 1) = int(Do - (B1 U $2)) isa fundamental domain for 0.

As we observed above, JD has two connected components; call these the innerand outer components, where the inner component is contained in H2.

The sides of Do on the inner component are all paired with one another,similarly for the sides of the outer component. However we chose f so thatthe side of W1 on the inner component is paired with the side of W2 on the outercomponent. This shows that Q(C)/?; has only one component. It is also clearthat D(d) has at least two components.

We need to show that D(G) has exactly two components. Use f to attach thetwo parts of 6 into a single connected fundamental domain B, C H2. We do thisso that the side pairing transformations of B, are those of Do together with thesesame generators conjugated by f. We likewise combine the two pieces of B toobtain a connected fundamental domain 12 contained in the outer component.We choose B2 so that its side pairing transformations are exactly the same asthose of 61.

Let H be the subgroup of t; generated by the side pairing transformations ofD1 (or of 62). We see from VII.D.6 that the relationship between H and 0 isindependent of 1'Y2. In particular, if we choose f2 = W2 as in E.6, then, since


the extended Fuchsian group has exactly two components, [t?i: H] = 2. SinceI is a fundamental domain for 0, and [0: H] = 2, 0 has exactly the twocomponents.

E.8. Crossed Fuchsian groupsFor any Kleinian group G, there is a natural subgroup G', called the derived

group, defined as the smallest subgroup of G containing every element thatstabilizes a component of G. Of course G' is normal in G, so A (G') = A(G).

If we choose n = 4, in the example in E.4, then the derived group G' is purelyloxodromic, and contains two Fuchsian subgroups, one of these subgroups keepsthe real axis invariant, and the other keeps the imaginary axis invariant; i.e., G'contains two Fuchsian groups whose limit circles intersect. If we choose n = 4m,then G' contains m distinct Fuchsian groups whose limit circles intersect at thesame two points.

E.9. A locally free Kleinian groupFor this construction, we start with a Fuchsian group of the second kind Go,

where B2/Go is a surface of genus 0 with one boundary component, and twoelliptic ramification points, both of order 3. Choose Go so that it has a funda-mental polygon Do, where Do is invariant under reflection in the real axis, andDo has one vertex at 0, as shown in Fig. VIII.E.14. Let h(z) = e2"tj3z, and let g bethe hyperbolic boundary element that identifies the sides s, and s; of Do; thenGo has the presentation <h, g : h3 = (h o g-')' = 1 >. Choose g so that s, lies in thefourth quadrant, and s; lies in the first.

Let e' be the point where s, intersects S'; then g(e") = e-". There are manychoices for Go, note that as t - 0, the fixed points of g tend to 1, and tr2(g) - 4;that is, g tends to a parabolic transformation.

Let S' be the arc of the circle passing through the fixed points of g, where S'makes an angle of n/4 with Si, and S' lies inside B2. Choose t sufficiently smallso that S' lies entirely within the sector {z I - n/3 < arg(z) < n/3}.

Let G, be the subgroup of Go generated by g, = g and g2 = h o g o h-'. No-tice that G, is purely hyperbolic, and free on the two generators g, and g2. Also

Fig. VIILE.14


Fig. VIII.E.15

Fig. V1II.E.16

W2/G, is a surface of genus 0 with three boundary components, all of the samegeodesic length (Fig. VIII.E.15 shows a fundamental domain for G, as a Kleiniangroup).

Let S, be the simple closed curve formed by S' and its reflection in St; notethat the outside angle of S, at the fixed points of g, is n/2. Let S2 = h(S,).

Let T' = h2(S'); define the arc D' of the circle passing through the fixed pointsof g3 = h2 o g o h-2 = (92 o g,)-t as follows.Plies between S' and the boundaryaxis of 93 in 132, and the inside angle between T' and 1' is ir/2. Let T, be thesimple closed curve formed by T' U D', and let T2 be the reflection of T, in Sl(S2 and T2 are very heavy in Fig. VIII.E.16, while S, and T, are merely heavy).

200 Viii. A Trip to the Zoo

Let f, be an element of M, where f, (T,) = S,; of necessity, f, maps the insideof T, onto the outside of S, , and f, conjugates <g, > onto <g, >. There are stilltwo essential choices for f,: either f, conjugates 93 into g, or into its inverse.Choose f1 so that f1 o 93 o f1 ' = g,. Let B, be the closed inside disc boundedby T,, and let B2 be the closed inside disc bounded by S,; then by VII.E.5,G2 = <G,, f,> is Kleinian, purely loxodromic, and G2 = G, *f,. Then G. hasthe presentation: <91,92,93,.ft :93092091 = f1 093ofi 1 091 ' = 1>.

Similarly, choose f2 to map the inside of T2 onto the outside of S2, but thistime so that f2 o 93 o f2 ' = g2'. The closed inside disc bounded by T. is notprecisely invariant under its stabilizer, but T2 is a strong block; the inside of T2and the inside of S2 are jointly f2-blocked, and the common exterior of T. andS2, together with the open inside of T2, and the open inside of S2, form a properinteractive triple. We can again apply VII.E.5 to conclude that G3 = <G2, f2) isKleinian and purely loxodromic, and that G3 = G2 * f,. A presentation for G3 is:

<91,92,93,f1,f2:93092/'091 =f1093o/'f '09((1''' =.20930.2'092 = 1)

_ <g3,f1,f2:93of2og3tof2 tof1 og3of1 t = I>.

Write t = f,, b = f2, and a = 93; then

G3 = <a,b,t: toa-' ot' = aoboa-' ob-'>, or

G3=<a,b,t:toaot' =boaob-'oa-'>.Let G be the subgroup of G3 generated by the elements {t'o a o t -'",

t'o b o t -'", m e l }. In a free group, only the identity lies in every member of thelower central series [84 pg. 159], and a = t -' o [b o a] o t = [t -' o b o t, t -' o a o t] =[t-'obot,[t-2o[boa]ot2] = is not theidentity in G. Hence G is not free.

We next show that G is locally free; that is, every finitely generated subgroupof G is free. Let H. be the subgroup of G generated by the elements {t' o a o t -'",t'"obot-"',IntI < k}. It is clear that every finitely generated subgroup of G iscontained in some Hk. Since every subgroup of a free group is free [51 pg. 95],it suffices to show that each Hk is free. Using the Reidermeister-Schreier method[51 pg. 86], it is easy to see that Hk is free on the generators {t-k o a o tk,t'" o b o t-I", I ml 5 k}. Hence G is locally free, but not free.

VIII.F. Strings of Beads

F.I. Let D be a fundamental polygon for the Fuchsian group G, acting on 32.Each side s of D lies on a circle orthogonal to §'; this circle determines ahyperbolic plane in IH3. The set of all these planes determines a polyhedronE c H'; it is easy to see that E is a fundamental polyhedron for G. Observe thatthe sides of E are paired by the same generators of G that pair the corresponding

VIII.F. Strings of Beads 201

Fig. VIII.F.1

sides of D. E has edges corresponding to the edges (actually vertices) of D, andE has no vertices in H 3; of course, E might have some infinite vertices at parabolicfixed points. This leads one to believe that one can deform the group by movingthe polyhedron, keeping the angles between the sides, and the Euclidean radiusof the sides, unchanged. In fact it is only in very special cases that one cannot dothis (the triangle groups). Most of the examples in this section are special casesof this procedure; for simplicity, we use groups generated by reflections, so thateach side of E is paired with itself.

F.2. Fuchsian reflection groupsLet C1, ..., C. be n circles, all orthogonal to S', where C. is tangent to both

C._, and and C, is a tangent to both C2 and C. (n = 8 in Fig. VIII.F.I).Let g,, be reflection in C.; extend g, to act on H3, so that g,, is reflection in thehyperbolic plane S. spanning C,,. The region E in H3, lying outside all the S., isa fundamental polyhedron for G = <g 1, ... , g, >. The only relations in G are thateach g,,, is an involution, and that g. o g,,+ t is parabolic (also, g, o gA is parabolic).

The points of tangency of the C. are parabolic fixed points in G. Everyparabolic fixed point x can be described by a symbol of the form { A,, ... , AR, ± },

where each A. is one of the circles: C1, ..., C,, and A. # A._1. This symbol hasthe following meaning. If k = 1, {A,, +} is the right endpoint of the circle A,,and {A 1, - } is the left endpoint of the circle A, (we define right and left so thatS2 lies to the right of S1). Note that {A 1, +) = {A2, - }. Fork > 1, x lies insideA,. Inside A,, there are translates of all the C. other that A1, x lies inside thetranslate of A2. Inside A2, there are translates of all the C. other that A2, x liesinside A3, etc. Finally, inside Ak_,, there are translates of all the C. other thanA,,_1, x is the right or left endpoint of the translate of A,,, according as the finalsymbol is + or -. The symbol (A,,..., A,,, ±) has length k - 1.


Every limit point x, other than a parabolic fixed point, can be described byan infinite symbol of the form { A,, A 2, ... }, where each A. is one of the circlesC,..... C", and A. A The symbol has the following meaning: x lies insideA and x lies inside the translate of A2 that lies immediately inside At (i.e., letf, denote reflection in A,, then x lies inside f,(A2)), etc. The symbol {A,.... } hasinfinite length.

These finite and infinite symbols can be used to reproduce the topology ofS'. A neighborhood of the symbol {A1,..., Am,... } consists of all points whosesymbols have length at least in, and which agree with the given one for at leastthe first m places. A neighborhood of {A,..... Ak, +} consists of all points cor-responding to the finite and infinite forms where Ak+llies immediately to the left of Ak, Ak+2 lies immediately to the left of A,+,, andso on up to Am, together with all points of the form {A,,..., At...... Am,... },where Ak. lies immediately to the right of At, Ak+

1lies immediately to the right

of A1., and so on up to Am.The important point of the topology introduced above is that it is indepen-

dent of the fact that all the C. are orthogonal to St. A string of beads is a set ofEuclidean spheres S,, ..., Sk, where each S. is orthogonal to 491-0", each S.lies outside all of the others, except that S, is tangent to S2, which is also tangentto S3, and so on to Sk, which is also tangent to S,. We reiterate the statementabove.

Proposition. Let S1, ..., Sk be a string of beads, let gm denote reflection in Sm, andlet G = <g 1, ... , gk >. Then A (G) is a topological circle.

The standard string of beads in H" is as above, with the additional hypothesesthat the S. all have the same Euclidean radius, and that the points of tangencyall lie on S'. It is clear that for a standard string of beads, the limit set is allof S'. The group generated by reflections in the standard string of k beads isdenoted by Rk.

F3. A quasifuchsian reflection groupThe most obvious use of the above is to deform the standard string of beads

inside the plane (see Fig. VIII.F.2), to obtain a quasifuchsian group. Actually, wehave to pass to the orientation preserving half, but this causes no difficulties.

F.4. A wild knotFor the next application, pick up the string of beads into 1=3, and cut it open

at a point of tangency; tie a knot in the string and close it up again, keeping allthe beads disjoint, except for the points of tangency. The knot keeps reproducingitself under the various reflections in the group, so in every neighborhood U ofa limit point, the limit set is infinitely knotted; i.e., U - A has infinitely generatedfundamental group. Hence A is a wildly embedded circle.

VIIi.F. Strings of Beads 203

Fig. VIII.F.2

Fig. VII1.F.3

F.S. A twisted Fuchsian groupStart again with a standard string of beads in H3. Pick up the string, put a

twist in it, and then lay it back down on El. Of course, we cannot really do thiswithout having some beads passing through others. However, we do this so thattwo of the beads coalesce; now there is one bead with four points of tangency,but otherwise, the beads have no additional points of contact (the "string" fromthe string of beads is shown in Fig. VIII.F.3 so as to help identify the figure; ithas no meaning in the context of Kleinian groups; see IX.I.12). The groupgenerated by reflections in all the beads is still discontinuous, and the outside ofall the beads is still a fundamental domain. There is a homomorphism from R.onto the group generated by reflections in this string, but this homomorphismhas no apparent geometric meaning.

F.6. A deformed Fuchsian groupFor this construction, start as above, with a standard string of beads, pick

up one of the beads into E3, pull it across some opposite beads and then lay itdown. We require that two of the beads coalesce, so that now there are two beadswith four points of tangency each, but otherwise, no new points of intersectionhave been introduced (n = 12 in Fig. VIII.F.4; as in Fig. VIII.F.3, the string isshown for the orientation of the reader, it has no significance for the Kleiniangroup). As above, there is a homomorphism (p: Rk -, G, where G is the group


Fig. VIII.F.4

generated by reflections in the beads of this string, but now there is also a localhomeomorphism f, from the outside of the unit disc, that realizes op. That is, forevery g e R,t, f- g = Mp(g) of One sees at once how to define f on the appropriatehalf of the fundamental domain for Rk; then use fog = co(g) of to define f onthe rest of the outside of the unit disc. In fact, the image of f is all of C, but thisis not obvious.

F.7. AtomsThe next construction is somewhat different. Let C and C' be two disjoint

spirals, both contained in the annulus (z I I < IzI < 2), where C and C bothwind infinitely often about both the unit circle and about the circle {IzI = 2).These two curves divide C into four regions: B2, (zI IzI > 2) U {oo), and the tworegions A and A' between C and C. Now construct an infinite sequence ofbeads, each bead tangent to both its neighbors on C, so that every point of Clies inside or on one of these beads. Similarly construct a sequence of beads onC so that every point of C lies inside or on one of these beads, and each beadis tangent to both its neighbors. We construct these two sequences of beads sothat except for the points of tangency mentioned above, the beads are alldisjoint.

Let G be the group generated by reflections in all the beads; we need to showthat D, the common outside of these beads, is a fundamental domain for G.Temporarily renormalize so that G acts on 133; the beads retain their essentialproperties under this conjugation. It is obvious that the bead itself is the isometricsphere for the reflection in the bead; hence the common outside of all the beadsin E3 is the Ford region. We saw in IV.G.6 that the intersection of the Ford regionwith I33 is a fundamental polyhedron; by VI.A.3, its boundary is a fundamentaldomain.

Let D c C be the fundamental domain defined by the exterior of all the beads.D has four components; these four components project onto distinct connectedcomponents of Q/G. Label these components as D, and D2, lying between thespirals, D3 = 032, and D4 = {zI IzI > 2} U ( oo). For m = 1, 2, let A. be the com-ponent of G containing.. Since every side pairing transformation is a reflection,A, and A2 are both G-invariant. Since they are both G-invariant, they are bothsimply connected.

VIII.G. Miscellaneous Examples 205

An atom is a component of a Kleinian group whose stabilizer is trivial. Wehave constructed a Kleinian group G, where Q(G) contains two simply connectedinvariant components, and also contains some atoms.

Our group G is infinitely generated. By Ahlfors' finiteness theorem, no finitelygenerated Kleinian group can have a disc as a connected component of DIG, soevery group with atoms is necessarily infinitely generated.

VIII.G. Miscellaneous Examples

G.I. Let G be a finitely generated Kleinian group; let N be the minimal numberof generators of G, let K be the number of connected components of 0/G. Oneexpects that K::5 2(N - 1), but this inequality is not known (at the time ofwriting, the best known inequality along these lines is that the number of com-ponents is bounded by 18(N - 1)). The first example in this section is a groupwith K = 2(N - 1).

Fix N > 1. For m = 0,..., N - 2, let C. be the circle {z I Iz + l - 2im)I = 11,and let C,, be the circle (zI Iz - I - 2im)I = l} (N = 4 in Fig. VIII.G.1). Letfm(z) = [(1 + 2im)z + 4m2]/[z + 1 - 2im]; observe that fm is parabolic withfixed point at 2im, and f,,, maps the outside of C. onto the inside of C.. Thecomputations below are most easily verified by observing that f, is the composi-tion of reflection in C. followed by reflection in the imaginary axis.

Let C,_, be the line {Re(z) = -2}, let CC_, be the line {Re(z) = 2), and letfN_t be the transformation, fN_1(z) = z + 4. Let Sm(S,,) be the hyperbolic planein H3 spanning QC.), and let E be the polyhedron bounded by the S,,, and SA,.

Fig. VIILG.1


Observe that fm(Sm) = S;,, and that f.(E) n E = 0. Since E has no edges, the onlyhypothesis of Poincare's theorem that we need to verify is the completenesscondition. There are 3(N - 1) + 2(N - 2) + I vertices at infinity as follows. Thetransformation fN't of. is parabolic with fixed point at -2 + 2mi; the trans-formation f,,, is parabolic with fixed point at 2mi; the transformation fm of;-2t isparabolic with fixed point at 2 + 2mi; the transformation fm+t of. is parabolicwith fixed point at -1 + (2m + 1)i; the transformation f.+1 of;` is parabolicwith fixed point at I + (2m + 1)i; and finally, fN_t is parabolic with fixed pointat oo.

We conclude from Poincarb's polyhedron theorem that E is a fundamentalpolyhedron for G = < fo, ... , fN _ t >. Since E has no edges, G is free on these Ngenerators; i.e., G = <fo) * <fN_l >. In particular, the minimal number ofgenerators for G is N.

Let D be the boundary at o0 of E; by VI.A.3, D is a fundamental domainfor G (in Fig. VIII.G.1, N = 4, and D is shaded). It is easy to see that D has3(N - 2) + 2 connected components. The components immediately to the rightof Re(z) = -2, and those immediately to the left of Re(z) = 2, are identified witheach other by fN_t. For each of the others, every side is paired with anotherside of the same component. Hence Q(G)/G has 2(N - 2) + 2 = 2(N - 1) con-nected components, and G is a free group on N generators. We also note thatevery maximal parabolic cyclic subgroup has rank 1, and that there are exactly2(N - 1) + (N - 2) + 1 = 3(N - 1) distinct conjugacy classes of such subgroups.

G.2. Let Go be the group constructed above, with N = 2, and let Eo be thefundamental polyhedron for Go. Let Et be the polyhedron formed by thefour sides of Eo together with the two planes {Im(z) = ± 1). Let g(z) = z + 24and let Gt = <Go, g)'. We can either use Poincarb's theorem as above, or wecan observe that {z I Im(z) 5 -1 } U (co) and {z I Im(z) 2t 1) U { oo) are jointlyg-blocked strong (<ft>,Go)-blocks, and then use VII.E.5 to conclude that Etis a fundamental polyhedron for G1. Then Dt = °aE1 is a fundamental domainfor Gt (see Fig. VIII.G.2).

It is easy to use the identifications of the sides to bring together all thecomponents of Dt lying over the same component of Q(G1)/G, (one possibility

Fig. VIII.G.2


Fig. VIII.G.3

for this is shown in Fig. VIII.G.3). One sees that £2(G, )/G, has two connectedcomponents, both surfaces of basic signature (0, 3; oo, oo, oo). Easy computationsshow that if G is a Kleinian group with two parabolic generators g, and 92,where the product g, 0 92 is also parabolic, then G is necessarily Fuchsian(normalize so that g,(z) = z + 1, and so that g2 has its fixed point at the origin,or see IX.C.2). We have shown that G, contains infinitely many Fuchsian sub-groups of the first kind. The boundaries of two of these components are shownin Fig. VIII.G.3.

Observe that G, contains a rank 2 Euclidean subgroup J = <g, f,) with fixedpoint at oo. Since every component of G, is a circular disc in the finite plane, noelement of J stabilizes any component of G,.

G.3. For the group G, constructed above, Q(G,)/G, has exactly two connectedcomponents; both are surfaces of signature (0, 3; oo, oo, oo). Also, every com-ponent of G, is a circular disc. Let B, and B2 be two inequivalent componentsof G,; f o r example, let B, = (z I I z + 2 - it 5 1), and let B 2 = {z I I z - it < 1) (seeFig. VIII.G.3). Let f(z) = (iz + 2i)/(z + 2 - i). Observe that f maps the points(-1 + i, - 2, - 3 + i, - 2 + 2i) onto the points (-1 + i, 0,1 + i, 2i), respectively,and that f is parabolic with fixed point at -1 + i. It is easy to see that f mapsthe outside of B, onto the inside of B2, and conjugates Stab(B,) onto Stab(B2).

Let J. be the stabilizer of B. in G1. Here again, (B1, B2) is not preciselyinvariant under but B, and B2 are both clean blocks, and are jointlyf-blocked. It is clear that there are limit points of G, that are not translates ofany point of B, U B2. We conclude from VII.E.5 that G = <G,, f> is discrete, butdoes not act discontinuously anywhere on C.

The group G constructed above is a subgroup of the Picard group, PSL(2,Z(i)), where Z(i) = {n + mi I n, m e Z} (in fact, G is a subgroup of finite index inthe Picard group).

GA Our next construction also yields groups that are discrete, but do not actdiscontinuously anywhere on C; the major difference being that here we constructgroups without parabolic elements.


Fig. VIIl.G.4

Fig VIII.G.5

We start with six circles, call them Ct,..., C6, where each of these circles isorthogonal to the unit circle, and each of them is orthogonal to two others, butotherwise there is no intersection among them. Using the fact that two circlesare orthogonal if and only if the square of the distance between their centersis equal to the sum of the squares of their radii, we can normalize so thatC. = {z j Iz - /2ezi Ij = 1) (see Figure VIII.G.4). Letf(z) = (f + V 2-)e"/6z,and let C;, = f(Cm), so that C, is a circle of radius (f + /2-) centered at.,,F2(f + T2)ex'(Zm*')/6. It is easy to verify that for each C. and C, eitherCm fl C = 0, or C. is orthogonal to Ck (see Fig. VIII.G.5). Let E be the polyhe-dron formed by the twelve (hyperbolic) planes spanning the C. and the C,.

Let fm denote reflection in Cm, and let f,,; denote reflection in C,,. Let F =<fi,...,f6>, F = and let G = Using Poincare's


polyhedron theorem, we see that G is a discrete subgroup of Ira. Also, F preservesthe disc A = {z I Izi < 11, and is a component subgroup of G. Similarly, F'preserves the disc A' = {z I Izl > f + f } U {oo}. Further, since c3E has onlythe two connected components, one in d and the other in A', every componentof G is a translate of either d or X.

Since the fundamental polyhedron E has no vertices at infinity, G containsno parabolic elements. Let H be the orientation preserving half of G; then H isKleinian; H contains no parabolic elements, but does contain elliptic elements;A and d' are non-equivalent components of H; and every component of H is atranslate of either A or A'. Also, J = Stabi(A) is a subgroup of index 2 in F, sois a geometrically finite Fuchsian group of the first kind. Similarly J' = Stab (d')is a geometrically finite Fuchsian group of the first kind. Since J and J' aredisjoint, G contains infinitely many distinct such subgroups; that is, G hasinfinitely many components, and each component is a circular disc. The distinctcomponents of G are disjoint; the content of the proposition below is that forsuch a group, distinct components have disjoint closures.

G.S. Proposition. Let H be a geometrically finite Kleinian group, where H containsno parabolic elements, H has more than two components, and every componentsubgroup of H is Fuchsian. Then distinct components of H have disjoint closures.

Suppose A and d' are distinct components of H, with stabilizer J and J',respectively, and suppose there is a point x c if n,4'. Since J and J' are Fuchsian,A and A' are necessarily circular discs; hence their closures can intersect in atmost one point. For any two Kleinian groups, it is clear that the limit set of theintersection is contained in the intersection of the limit sets; since H contains noparabolics, J fl J' is finite.

Since H is geometrically finite, it is analytically finite; hence J is also analyti-cally finite; then by VLE.4, J is a geometrically finite Fuchsian group of the firstkind. Hence x is a point of approximation for J; i.e., there is a sequence of distinctelements { jm} in J, with jm(x) - x', and jm(z) -+.y' # x', for all z 0 x. Since noloxodromic element of G can be in both J and J', it follows from II.C.6 that eitherx is not a fixed point of any element of J, or x is not a fixed point of any elementof X. Interchanging A and A' if necessary, we can assume that x is not fixed byany jm. Since jm(x) is the point of intersection of A and jm(A'), the sets areall distinct. The discs j. (.d') converge to a set passing through both x' and y',hence a disc.

On the other hand, the sets jm(d') are all distinct; as in VII.B.9, normalize sothat oo a °Q(G), and replace each jm by a transformation of the formjm o wherej;,, a J' = Stab(A'), and the center am of the isometric circle of jm o j' lies in aconstrained fundamental set for X. Since am is bounded away from d', and theradii of the isometric circles converge to zero, dia(jm(A')) - 0. This contradictsthe statement above that these translates of A' converge to a disc. 0


G.6. We continue with the construction of G.3. While J and J' are not conjugatein H, they are conjugate in M; in fact, the transformation f conjugates J into J';it also maps d onto the exterior of d'. It is easy to see that there must be limitpoints of G that do not lie on any translate of either ad or ad'. Then by VII.E.5,K = (H, g> is discrete, geometrically finite, has no parabolic elements, and doesnot act discontinuously anywhere on C. It follows that any fundamental polyhe-dron for G is relatively compact in H3.

Observe that we could replace f by any of the five transformations: z -(,vF3 + m = 1, ..., 5. Each such transformation maps d onto theexterior of

A'and conjugates J onto Y. These six choices for f yield six distinct

(i.e., non-conjugate in M) groups. These six distinct groups all have the samefundamental polyhedron in 0-03, with different identifications.

G.7. It is fairly straightforward to find a subgroup H' of the group H constructedabove, where H' has finite index in H, H' is purely loxodromic, and the stabilizersof d and d' in H' are still conjugates of each other under any of the six elementslisted above. Then we can again use VII.E.5, to form a group K', which is of finiteindex in K, and hence also geometrically finite and of the first kind (i.e., does notact discontinuously anywhere on (C), where K' is purely loxodromic.

VIII.H. Exercises

H.I. Let L be the geodesic in H2 defined by the positive imaginary axis. Let Wbe the set of points whose hyperbolic distance from L is some positive constantd. Show that W consists of two Euclidean rays emanating from the origin, andmaking equal angles with L. Relate this angle with the hyperbolic distance d.

H.2. Construct a sequence of pairs of circles {Cm, C;, } bounding a common regionD, and corresponding elements { gm } of M, with gm(CC) = C;,, and gm(D) n D = 0,so that G = <91, 92, ... > is Kleinian, but D is not a fundamental domain for G.(Hint: Look at the polyhedron in H3 bounded by the spanning discs for the C.and C,,. Construct these so that the identified polyhedron is not complete. Usecombination theorems to show that G is Kleinian and that D is preciselyinvariant under the identity in G. Since the identified polyhedron is not complete,there is a sequence {gk(C,)} of translates of sides of D so that dia(gk(C )) doesnot converge the zero.)

H.3. Letj(z) = z + 1. Construct a Kleinian group G containing j, where StabG(oo)has rank 1, but oo is not doubly cusped in G.

HA. Construct a Kleinian group G, and a closed set B, so that B is preciselyinvariant under the geometrically finite subgroup J of G, but B is not a (J, G)-block.

V1II.H. Exercises 211

H.5. Construct a Kleinian group G that is not geometrically finite, where oo e°Q(G), and for every limit point x, other than parabolic fixed points, there is agEG, with (g'(x)l > 1.

H.6. Let S be a marked finite Riemann surface of signature a = (p, n; v,,..., vp),where p > 0. Let w be a simple loop on °S, where either w is non-dividing, or wdivides °S into two surfaces, each of which either has positive genus or at leastthree boundary components. Show that there is a Fuchsian group F of signaturea, with topological projection map p: H2 - S, so that the lift of w determines ahyperbolic element of F.

M. Given the number a > 0, show that there exists a Fuchsian group G of thesecond kind, where 82/G is a torus with one hole, and any primitive boundaryelement of G has multiplier a. (Hint: Let G be generated by a and b, where a andb are hyperbolic, a has its fixed points at ± 1, b has its fixed points at ±i, andthe isometric circles of a and b have equal radii.)

H.8. Given three positive numbers /i,, fl2, fl3, show that there are three hyper-bolic lines L1, L2, L3 in H2, all bounding a common region in H2, so that!'1 = d(L2, L3), #2 = d(L3, L1), 03 = d (L, L2). (Hint: use H.1.)

H.9. Given three numbers a a2, a3, all greater than 1, show that there is aFuchsian group of the second kind, representing a sphere with three holes, sothat the distinct primitive boundary elements have multipliers a,, a2, and a3.(Hint: Use H.8 with ft. = and consider the group generated by thereflections in L1, L2, and L3.)

H.10. Let n = 2p + k - I > 1 be given. Show that for any choice of numbersa ..., ak, all greater than 1, there is a Fuchsian group G of the second kind,where B2/G has genus p, with k holes, and the k distinct primitive boundaryelements have multipliers a ..., ak. (Hint: Use induction on k. For k = 1, andp > 0, generalize the construction of H.7; for k = 3, and p = 0, use H.9. For theinduction step, construct a group G', on n - I generators, where the boundaryelements G' have multipliers a,_., ak_2, a;_1.)

H.11. Construct two geometrically finite Kleinian groups G and 6, so that thereis a type-preserving isomorphism cp: G -+ 6, but there is no homeomorphism0: Q(G) -,12(0) conjugating G into Cy. (Hint: Choose these groups to be freeproducts of a Fuchsian group of the first kind with a loxodromic cyclic group.)

H.12. Construct a Kleinian group G, containing an elementary subgroup J ofthe form 7 + Z,,, where no element of J stabilizes any component of G.

H.13. Construct a Kleinian group G containing a double dihedral group J, whereno element of J stabilizes any component of G.

212 Vlll. ATriptotheZoo

H.14. Construct a Kleinian group G, having a component d, where the numberof boundary components of d is greater than two, but finite (that is; rct(A) isfinitely generated and not cyclic).

H.15. Let G be the group constructed in G.3. Show that G is of finite index inPSI (2,1(i)). What is the index (see IV.J.22)?

H.16. Let d be a component of the Kleinian group G, where d is a strong(Stab(d), G)-block. Let {gm(A)} be an enumeration of the distinct G-translates ofA. Prove that Y oo.

VIII.I. Notes

There is a large collection of examples in Krushkal', Apanasov, and Gusevskii[46]; several of the examples presented here can also be found in their collection.The interested reader can find an account of the theory of spaces of Kleiniangroups in Bers [16]. A.I. The existence of non-classical Schottky groups wasshown by Marden [53]; a specific example was constructed by Zarrow [103].B.I. The term "simultaneous uniformization" was first used by Bers [13] whoshowed that given two finite marked hyperbolic Riemann surfaces Sl and S2 ofthe same signature, there is a quasifuchsian group G so that S2/G is the disjointunion of St and S2. C.4. Limit points that do not lie on the boundary of anycomponent are called residual limit points; these were explored by Abikoff [1],and others. E.2. The operation on the underlying surface corresponding to slidingis sometimes called a partial Dehn twist, or a Fenchel-Nielsen twist; see Wolpert[100]. E.3. The concept of "bending", which involves hyperbolic surfaces in H3,was introduced by Thurston [90]; for quasifuchsian groups, the operation ofbending as described here is closely related. EX This group was first constructedin [66]. E.9. Thanks are due to Gilbert Baumslag both for the exact form of thisgroup, and for the proof that it is locally free, but not free. Another example ofa locally free but not free Kleinian group can be found in [69]. F.3. This examplefirst appeared in Fricke-Klein [26 pg. 415-418], and has been rediscovered manytimes. F.4. This example was discovered independently by several people; see[10]. F.6. A deformation of a Fuchsian group F is a locally univalent functionf(z), defined on H2, together with a homomorphism cp: F - PSL(2,C), wherefog = <p(g) of for all g e F. These have been extensively studied by Gunning [30],Kra [42], and others. See [42] for a proof of the fact that if f is not a cover map,then the image of f is all of C. F.7. Accola discovered atoms using this example[3]. G.I. Bers [14] proved that K 5 84(N - I); Ahlfors [7] improved this toK < 18(N - 1). The inequality K < 2(N - 1) has been proven in various specialcases by Marden [52], Nakada [77], and Abikoff [2]. See Sullivan [88], alsoAbikoff [2] and Kra [44], for recent work relating the maximal number of

VIII.]. Notcs 213

incongruent parabolic fixed points with the minimal number of generators. G.2.There is a similar example in Fricke-Klein [26 pg. 428-432] (see especially thefigure on pg. 432). G.6. Wielenberg [98] discovered a fundamental polyhedronfor the Picard group that can also serve as a fundamental polyhedron for othergroups, using different identifications. H.14. Kalme [38]. H.16. [66].

Chapter IX. B-Groups

A B-group is an analytically finite Kleinian group with a simply connected in-variant component. We divide the non-elementary B-groups into three classes:Fuchsian and quasifuchsian groups (these are groups with exactly two com-ponents, both invariant), degenerate groups (these are groups with exactly onecomponent), and groups that contain accidental parabolic transformations. Wealso demonstrate the existence of degenerate groups, and discuss the structureof groups containing accidental parabolic transformations.

IX.A. An Inequality

A.I. A conformal metric on a Riemann surface S is a form i which transforms sothat i.(z)Idzl is invariant, and so that in every local coordinate A(z) is a positivesquare-integrable function. The natural metric on S is obtained by going upto the universal covering surface X, and projecting the spherical, Euclidean,or hyperbolic metric from X; i.e., if p: X - S is the universal covering, and;.I(z) is the standard metric on X, then the natural metric on S is defined by%(p(z))Ip'(z)1 = 1.*(z).

If S is a plane domain, then a conformal metric on S is simply a positive L2function.

For any Riemann surface S, and any conformal metric d on S, the A-area ofS is

AA(S) = 1/2 fS

)2(z)Idz A dil.

If w is a rectifiable curve on S, and A is a conformal metric, where A isintegrable along w, then the A-length of w is JW A(z)ldzl. If E' is a family of rectifiablecurves on S, then the A-length of the family is given by

A(z)ldzl.LA(Z,S) = inf fWWEE

The extremal length m(E, S) of the family of curves £ on S, is defined bym(E, S) = sup(L,,(I, S))2/AA(S), where the supremum is taken over all conformalmetrics A, for which A,, 0 0.

IX.A. An Inequality 215

One of the usual families of curves for these calculations is the family of allloops freely homotopic to a given one. If w is a loop on S, then m(w, S) = m(E, S),where E is the family of loops freely homotopic to w on S.

A.2. There are only a few special cases where one can actually compute the ex-tremal length. One of these is the family of curves which separate the boundariesof an annulus.

Let R be the annulus R = {zl I < Izi < t}, and let S = {zI0 <Re(z) < logt}.The group of translations G = {z - z + 2nni, n e l } acts on S so that ez: S --+ Ris the universal covering. A conformal metric on R is then a G-invariant conformalmetric on S, and a loop which separates the boundaries of R is a G-invariantcurve in S. Let D be the rectangle D = {zI0 < Re(z) < log(t), 0 < Im(z) < 2n}; Dis a fundamental domain for the action of G on S.

Let A be a G-invariant conformal metric on R, where Ax(RIG) = A,(D) < oc.For every x with 0 < x < log(t), the vertical line passing through x is a curve inour family. Hence, using the Schwarz inequality,

(IX.1)

2

(Lx)2 5(f,*

i.(x + iy)dy)-

2"

< 2n A2(x + iy)dy.0

Integrate the above inequality from 0 to log(t) to obtainlog(r) 2x

(Lx)2log(t) < 2n A2(x + iy)dydx0 o

5 2nA t.

This yields (Lt)2/Ax 5 27/log(t). Of course the Euclidean metric in S yieldsequality. Hence m(A, S) = 2n/log(t), the usual modulus of the annulus R.

A.3. Let w be a simple loop on the Riemann surface S, and let R be a subsurfaceof S, with w c R c S, where R is a topological annulus, and w separates theboundaries of R. Every conformal metric on S can be restricted to a conformalmetric on R, and every conformal metric on R can be extended to a conformalmetric on S. Every loop on R freely homotopic to w can also be regarded as aloop on S freely homotopic to w. Hence m(w, R) >_ m(w, S).

Set m(w, S) = inf m(w', R'), where the infimum is taken over all topologicalannuli R', where w' is freely homotopic to w, and w' c R' c S.

AA. Let G be a Kleinian group, and let A be a component of G. Set H = Stab(A),let S = d/H, and let w be a simple loop on °S, where the element of H corre-sponding to w is loxodromic; that is, there is a loxodromic element g e H, andthere is a g-invariant curve W, where W projects onto w. Normalize so that

216 IX. B-Groups

g(z) = k2z, I k I > 1, and so that the point I lies on W. Define log(k) by analyticcontinuation along W, where log(l) = 0.

Proposition. Ilog(k)I < nth(w, S).

Proof. Let w' be any loop on S freely homotopic to w, and let R' be any annuluswith w' R' c S. Let Y be the connected component of p- (R') containing W.Let f (z) = log(z), defined in Y, where, as above, log(I) = 0. Let i be the Euclideanmetric in f(Y). Then ;. is a conformal metric on R'.

A loop v, freely homotic to w, appears in f(Y) as a path whose endpoints areconnected by the transformation z -+ z + 2log(k); the A length of such a path isat least 2Ilog(k)I. Since f = log(z) is univalent in Y, f(Y) intersects each verticalline in a set of linear measure at most 2n; hence A;,(R') 5 4n(Re(log(k))). Weconclude that

m(w', R') >_ 41log' (k)I/4n Re(log(k)) Ilog(k)I/n.

Hence Ilog(k)I < nm(w', R'), for all possible w' and R'; the desired result followsby taking the infimum. 0AS. Corollary (Ahlfors [4]). Let G be a Kleinian group and let S be a connectedcomponent of DIG. Suppose there is a punctured disc R conformally embedded inS. Let g be an element of G corresponding to a small loop about the puncture. Theng is not loxodromic.

A.6. Corollary (Yamamoto [101 ] ). Let { Gm } he sequence of Kleinian groups. whereS2(G,n)/Gm contains the surface Sm. Let w be a simple loop on So, where w cor-responds, under the natural homomorphism, to a loxodromic element of Go. Supposethat for every m, there is a homeomorphism fm: So - Sm, so that f(w) = isrepresented in G. by a loxodromic element gm. If m(wm, Sm) - 0, then the multiplierof gm converges to 1.

IX.B. Similarities

B.I. A function group (G, A) is a Kleinian group G, with an invariant componentA, where 4/G is a finite Riemann surface. We will see below that every functiongroup is analytically finite; that is, S2/G is a finite (disconnected) Riemann surface.We will often refer to the function group (G, A) as simply G; this should cause noconfusion.

A similarity between function groups, (G, d) and (C, d ), is a homeomorphismf : A j, with induced isomorphism f,: G - (, so that f o g (z) = f,(g) of(z) forall g c- G and for all z c- A. The similarity f is conformal if the mapping f isconformal. Unless specifically stated otherwise, all similarities are assumed to beorientation preserving.

IX.C. Rigidity of Triangle Groups 217

Note that the isomorphism induced by a similarity need not be type-preserving;that is, it might take a loxodromic element into a parabolic. This is in contrastto a deformation, where the induced isomorphism is always type-preserving. Atype-preserving similarity is one for which the induced isomorphism is type-preserving.

We have required a similarity to be only continuous. Since every homeomor-phism of a Riemann surface can be approximated by a diffeomorphism, we willusually assume that all similarities are differentiable (in the context of Riemannsurfaces, it is more natural to assume that all similarities are quasiconformal; un-fortunately, the study of quasiconformal deformations and similarities is beyondthe scope of this book).

B.2. A B-group is a function group (G, A), where A is simply connected.A B-group with exactly two components, both invariant, is called a quasi-

fuchsian group. This definition is seemingly weaker than the one given in VIII.E.3;the two definitions are however equivalent (see F.10).

An elementary B-group has exactly one component, obviously invariant, andthis one component either is the sphere, or is conformally equivalent to the plane(i.e., the sphere with one puncture).

If (G, A) is a non-elementary B-group, then c'A has more than one point, soA is conformally equivalent to the disc. There is a Riemann map #: .4 -i B2; henceevery non-elementary B-group is conformally similar to a finitely generatedFuchsian group F = fGf`; This Fuchsian group F is called the Fuchsian modelof G. A Fuchsian group represents a finite Riemann surface if and only if it isfinitely generated and of the first kind. Hence the Fuchsian model of a B-groupis always finitely generated and of the first kind.

B.3. A non-elementary B-group (G, A) with exactly one component (i.e., A = 0)is called a degenerate group.

IX.C. Rigidity of Triangle Groups

C.I. The main object of this section is to show that there are no non-trivial de-formations of the Fuchsian triangle groups; more precisely, we will prove thefollowing.

Theorem. Let F be a Fuchsian group with signature (0, 3; v1, v2, v3), and let gyp: F - Gbe a type-preserving similarity onto a Kleinian group. Then there is an elementgeFA sothat (p,1,(f)=gofog',forall feF.

Proof. Assume that we are given a Fuchsian triangle group F, of signature(0, 3; v1, v2, v3), vl < v2 < v3, and assume that there is a type-preserving similarity(p:F-+ G.

218 IX. B-Groups

Choose generators a and b for G, so that a has order v1, b has order v2, andboa has order v3.

C.2. If v, = v2 = v3 = oo, then normalize so that the fixed point of a is at 0, thefixed point of b is at oc, and the fixed point of boa is at 1. In H3, draw the(hyperbolic) lines L, connecting 0 and oo, L. connecting co to 1 and L3 con-necting 1 to 0. Let r,. be the half-turn about L.. We know that b can be writtenas r o r,, and that a can be written as r, o F, where r is a half-turn about some lineL with one endpoint at oo, F is a half-turn about some line Lwith one endpointat 0. We also know from V.B.5 that boa is parabolic with fixed point 1 if andonly if the lines L and Lboth have their second endpoint at 1. This means thatL = L2 and L = L3. We have shown that a and b are completely determined bythe signature and normalization.

C.3. For all the other cases, we first establish that G is necessarily Fuchsian.Assume that v, < oo, and v2 = v3 = oo. Since F is non-elementary and Fuch-

sian, G is non-elementary (V.G.6), hence a and b cannot share a fixed point. Wenormalize so that the fixed points of a are at 0 and oo, and so that the fixed pointof b is at 1. Let A be the axis of a in 0-03, and let L2 be the line orthogonal to Awith one endpoint at 1; then the other endpoint is at -1. Let r2 be the half-turnabout L2. Write a = r2 o r where r, is a half-turn about a line L1, orthogonalto A, and passing through the point of intersection of A and L2. Both endpointsof L, lie on S'. Since r2 has one of its endpoints at 1, we can write b = r3 o r2,where r3 is the half-turn about a line L3 with one endpoint at 1. In order forboa = r3 o r, to be parabolic, L3 and L, must have a common endpoint; that is,the other endpoint of L3 is at one of the endpoints of L both of which lie on5'. Since both endpoints of L3 are on 5', r3 preserves S' (it is obvious that ahalf-turn with one fixed point at oc, and the other on the real axis, preserves thereal axis). Since r2 and r3 both preserve 5', and interchange the two discs itbounds, b = r3 o r2 preserves B2. We have shown in this case that G is Fuchsian.

C.4. We next consider the case that v, and v2 are both finite, and normalize asfollows. Let A be the axis of a, and let B be the axis of b. Since G is non-elementary,A and B do not intersect, even at the sphere at infinity. Normalize so that thecommon perpendicular of A and B has its endpoints at 0 and oo, with 0 beingcloser to A, and so that one of the fixed points of a is at 1. Then the other fixedpoint of a is at - I, and the two fixed points of b are of the form ±z, with Izl > 1.

Let L, be the line in H3 from 0 to oo; write a = r, o r3 and b = r2 or,, wherer2 is a half-turn about a line L2, orthogonal to B, and passing through the pointof intersection of B with L,, and r3 is a half-turn about a line L3 orthogonal toA, and passing through the point of intersection of A with L,. Note that sinceL3 is orthogonal to A, its endpoints on the sphere at infinity lie on the imaginaryaxis. Since boa is either elliptic or parabolic, L3 and L2 meet either in H3 or onthe sphere at infinity. Let P3 be the hyperbolic plane spanning the imaginary

IX.C. Rigidity of Triangle Groups 219

axis; note that P3 is orthogonal to A. Let P2 be the plane orthogonal to B andpassing through Lt. The line L3 lies in P3, and the line L2 lies in P2. The pointof intersection of these two lines lies in the closure of both planes; hence bothplanes contain Lt and a point not on (the closure of) Lt; it follows that the twoplanes coincide. Since B is orthogonal to P2 = P3, its endpoints are both real; itfollows that b preserves the right half-plane. Since a has fixed points at ± 1, italso preserves the right half-plane.

C.5. Now that we know that G is Fuchsian, we can use 2-dimensional hyperbolicgeometry. Normalize so that the fixed point of a in B2 is at the origin, and sothat the fixed point of b is real and positive. Let L be the line which connects orpasses through the fixed points of a and b. Let M be a line passing through, orending at, the fixed point of a, so that a can be written as first reflect in L, thenreflect in M. Similarly, let N be the line through, or ending at, the fixed point ofb, so that b can be written as first reflect in N, then reflect in L. Then boa is thecomposition of first reflect in N, then reflect in M. Since boa is either elliptic orparabolic, M and N meet either in H2, or on the sphere at infinity.

Let M' = a-' (M), and let N' = b(N). If M = M' (i.e., vt = 2), then there isessentially only one possibility for the lines M = M', N, and N'; this is shown inFig. IX.C.1, where we have assumed v3 = oo. The only other possibility for thiscase is that we could interchange N and N'; this can be accomplished by con-jugating G by the transformation z - -z.

If vt > 2, there are two possibilities. We decompose M into two half-lines:Mt in the right half-plane, and M2 in the left. If necessary, conjugate by z - 11zso that Mt lies in the first quadrant. Likewise decompose N into two half-linesjoined at the fixed point of b: N1, in the upper half-plane, and N2 in the lower. IfN, meets M1, the picture is as shown in Fig. IX.C.2, while if N2 meets M2, thepicture is as shown in Fig. IX.C.3 (in both cases, we have again assumed thatv3 = oo). In the first case, where Mt meets Nt, let Gt be the group generated bya and b; in the second case, call the group G2.

Notice that for Gt, M and N' are disjoint, so that b't o a is hyperbolic, whilein G2i M and N' intersect in 32, so that b-t o a is elliptic. This shows that thenatural correspondence between Gt and G2, given by the fact that they are both

Fig. IX.C.t

220 IX. B-Groups

Fig. IX.C.2

Fig. IX.C.3

generated by elements called "a" and "b", is not an isomorphism. In either case,if v, < oo, then M and M' meet at an angle of 2n/v,; similarly, if b is elliptic, thenN and N' meet at an angle of 2it/v2; and if v3 < oo, then M and N meet at thefixed point of boa at an angle of n/v3.

Consider G1. Let D be the polygon determined by M, N, M', and N'. ByPoincarb's polygon theorem, G, is discrete, D is a fundamental polygon for G1,and G, has basic signature (0, 3; v1, v2, v3). Observe that D is composed of twotriangles. One of these triangles is formed by the sides L, M, and N, having anglesof n/v,, a/v2, and 7C/v3. The other triangle is formed by the sides L, M', and N',and has the same angles. In hyperbolic geometry, a triangle is uniquely determinedby its angles, so these triangles, and hence D, are completely determined by thenumbers v v2, and v3, together with our normalization, and the assumption thatM, meets N,. In particular, G, is conjugate to F. Since b-1 o a is hyperbolic inboth F and G,, but not in G2, G = G,. 0

IX.D. B-Group Basics

D.I. Proposition. Let G be a non-elementary Kleinian group with an invariantcomponent J. Then 8d = d (G).

Proof. Pick a point z e d, and let g be a loxodromic element of G. Then the iteratesof z under both positive and negative powers of g converge to the fixed points

IX.D. B-Group Basics 221

of g. Hence both fixed points of g lie on the boundary of A. Since the loxodromicfixed points are dense in the limit set (see V.E.3), 4(G) = dd.

D.2. Proposition. Let G be a Kleinian group with an invariant component A. Thenevery other component of G is simply connected.

Proof. Every other component is a connected component of the complement ofthe connected set d.

D.3. Proposition. Let d he a component of the Kleinian group G, and let H =Stab(d). If d/H is a finite surface, then d is a component of H.

Proof. The limit set of H is contained in the limit set of G; hence there is acomponent d of H containing A. Then S = 4/H is conformally embedded inS = L/H. Since S is a closed surface with a finite number of points removed,S - S consists of at most a finite number of points. It follows that d - A consistsof at most a discrete set of points. Since A(H) is perfect, no point of d - d liesin A (H); hence d = A.

D.4. Proposition. Let G bean analytically finite Kleinian group, and let gyp: Q(G)Q(G) he a homeomorphism that commutes with every element of G. Then there isa deformation cp, of G onto itself, so that c0 I0 = cp, and 01 4 = 1.

Proof. Since G is analytically finite, it has a fundamental domain D with thefollowing properties. D has finitely many connected components; there is a finiteset of cusped regions B,, ..., BB, at parabolic fixed points of G, where each B.has non-empty intersection with D; for each m, D n B. is a cusp with vertex atthe center of B,,,; except for these cusps, D is relatively compact in Q.

Since cp commutes with every parabolic element fixing the center xm of thecusped region Bm, cp(Bm) is also a cusped region at xm. This statement remainstrue as we let B. shrink; hence we can continuously extend from inside B., tobe the identity at xm.

Let {zm} be a sequence of points of S2 with Z. - zE A. For each m thereis a point xm e D, and there is an element gm E G so that zm = gm(xm). Thencp(zm) = cp o gm(xm) = gm o cp(xm). Clearly we can assume that the gm are all distinct.

Choose a subsequence so that xm - x c -D. There are two cases to consider;either x e 0, or x is a parabolic fixed point.

If x e 9, choose a nice neighborhood U about x. Then dia(gm(U)) - 0, sogm(x) -+ z; hence gm(C,) -. z for all i; e0. In particular, gm((p(x)) -+ z.

If x is a parabolic fixed point, then cp(xm) - x from within some B;, so thepoint gm o V(xm) lies within gm o cp(B;). If the gm all represent the same left cosetof Stab(x), then write gm = g o hm, hm e Stab(x). Then the points zm = gm(xm) alllie within g(Bj), so z = g(x). Similarly, the points gm o cp(xm) = g o cp o hm(xm) all liewithin g o cp(B;), so cp(zm) = gm o (p(xm) - g(x) = z.

222 IX. B-Groups

If the elements gm all represent distinct left cosets of Stab(x), then we use thefact that the closure of a cusped region is a block. By VII.B. 14, dia(gm(B)) -+ 0for any cusped region centered at x; hence dia(gm o q (B;)) - 0. Since g(xm) -+ z,gm(Bj) - z. Hence gm o p(C) -+ z for all ( e Q ; so ,9m o cp(B;) - z. It follows thatp(Zm) gm 0 0(xm) - Z.

D.5. Proposition (Accola [3]). Let G be an analytically finite Kleinian group withtwo invariant components d, and d2. Then Q(G) = dl Ud2.

Proof. Suppose there were a third component d3. By D.3, H = Stab(d3) isnon-elementary. Let g be a loxodromic element of H, with fixed points x and y.Then in dm, there is a g-invariant path W. connecting x to y. These three pathsseparate t into three regions, call them R R2, and R3, where W, does not lieon OR.. Except for x and y, the closure of A. does not intersect Rm. Sincead, = A (G), R, is devoid of limit points. This contradicts the fact that W2 andW3 lie in different components of G.

D.6. Let (G, d) be a function group, and let j be a parabolic element of G. Supposethere is a conformal similarity f from (G, d) onto another function group (0,J),where f,(j) is not parabolic. Then j is called an accidental parabolic element of G.

It is important to remark that every Abelian subgroup of a Fuchsian grouphas rank 1; hence every purely parabolic subgroup of a non-elementary B-grouphas rank 1.

D.7. Proposition. A Fuchsian group of the first kind contains no accidental parabolictransformations.

Proof. Let j be a primitive parabolic element of F with fixed point x. Then (VI.A.7)there is a punctured disc U conformally embedded in o-u2/F so that the boundaryof U corresponds to j. By Ahlfors' lemma (A.5), there cannot be a conformalsimilarity f from (F,142) onto some (G, d) where f,(j) is loxodromic.

D.8. Proposition. Let (G, A) be a non-elementary B-group, and let j be a parabolicelement of G. Let fl: d -+ 142 be a Riemann map. Then j is accidental if and onlyif f,(j) is hyperbolic.

Proof. If f,(j) is hyperbolic, then of course, j is accidental. If I,(j) is parabolic,then, by D.7, &(j) is not accidental; hencej is not accidental.

D.9. Since a conformal similarity takes an elementary group to a group that isagain elementary, with the same number of limit points, no parabolic element ofan elementary group can be accidental.

D.10. Let (G, d) be a non-elementary B-group, and let j be an accidental parabolicelement of G. Let /3: d - fl2 be a Riemann map. We define the axis and true axis


of j to be the inverse image under fi of the axis and true axis, respectively, of fl ,(j)(see V.G.11).

The axis A of j is j-invariant. If we start with a pointy on A, then the iteratesof y under both positive and negative powers of j converge to the fixed point xof j. We adjoin x to A, to obtain a simple closed j-invariant curve. From here onwe consider this simple closed curve, with a distinguished point on it, to be theaxis of J. Since the set of axes in the Fuchsian model F is F-invariant, the set ofaxes of accidental parabolic transformations in G is G-invariant.

D.11. We make the following observations about axes of accidental parabolictransformations. If A, A' is the axis of the accidental parabolic transformation j,f, respectively, then it is not quite true that A = A' if and only if <j> = <j'>. Ifthe axes are also the true axes, then this statement is true. However if A is notthe true axis of j, then there is a half-turn h, with fixed points on the true axisof j, so that Stab(A) = Stab(h(A)) = <j>. The two axes bound a strip (if the fixedpoint of j is at oc, then this strip is bounded by two parallel curves). The strip isprecisely invariant under the stabilizer of the true axis, and contains the true axis.

D.12. Proposition. Let J = <f> he a maximal parabolic cyclic subgroup of thenon-elementary B-group G. J is not accidental if and only if there is a preciselyinvariant simple loop W, so that, except for the fixed point of J, W is containedin °d = °o fl A, and one of the open discs bounded by W contains no limit pointsof G.

Proof. Let #:,d -* 0-82 be a Riemann map, let F = /f#(G) be the Fuchsian model,and letj = ft*(j). Then j is accidental if and only if j is hyperbolic. If W is anyprecisely invariant curve in °d, then IV = /1(W) is precisely invariant under

If J is not accidental, then 3 is parabolic, and if we adjoin the fixed point ofto 1', we obtain a simple closed curve. One of the discs bounded by iP is

contained in 0-82; since #-' is well defined on this disc, W bounds a disc thatcontains no limit points.

Conversely, if J is accidental, then .7 is hyperbolic. Let W be any preciselyinvariant loop which, except for the fixed point of J, is contained in A. Since 7is hyperbolic, there are limit points on both sides of i$'; in particular, there areinfinitely many translates of Y ' on either side of IV. Choose translates W, and YP2on either side of I$', where Stab(W,,) does not commute with Stab(I'). Then theloops W. = lie on either side of W. Since Stab(Wm) does not commutewith Stab(W), the parabolic fixed point on W. is also disjoint from W; hencethere are limit points of G on either side of W. 0

D.13. Corollary. If there are B-groups (G, A) and (6, J), where G c l;, and J d,and j is an accidental parabolic transformation of G, then j is accidental in t;.

224 1X. B-Groups

D.14. Proposition. Let j and j' be non-commuting accidental parabolic transfor-mations of the B-group (G, A). Let A, A' be the axis of j, j', respectively. ThenAn A'=0.

Proof. Let fl: d - H2 be a Riemann map. Since the axes, f(A) and f(A'), intersectin at most one point of B2, A and A' intersect in at most one point of A. Twosimple closed curves on the sphere intersect in at least two points; the secondpoint of intersection of A and A' can only be the fixed point of both j and j'. Sincej and j' do not commute, this cannot occur.

D.15. Proposition. Let j be a primitive parabolic element of the non-elementaryB-group (G, d), and let x be the fixed point of j. Suppose there is a j-invariant simpleclosed curve W, with the following properties. W e A U {x}; W fl A is preciselyinvariant under J = <j> in G; and one of the open discs B bounded by W containsno limit points of G. Then B is precisely invariant under J in G; in particular, j isnot accidental.

Proof. Except for the fixed point of J, W lies in d, and W fl d is precisely invariantunder J; hence no translate of W intersects W (of course, there might be someg E G so that g(W) and W intersect exactly at the fixed point of J, but in this case,the two loops do not cross). Also, since B contains no limit points, no translateof W lies inside B. Hence the only possibility for a translate g(B) to intersect Bis to have B contained in g(B). Since B contains no translate of W, it containsno translate of itself. Hence B does not lie in any translate of itself. We haveshown that B is precisely invariant under J in G. By D.12, j is not accidental.

O

D.16. Corollary. Let j be a primitive accidental parabolic transformation in theB-group (G, d). If J = Q> is doubly cusped, then both cusped regions lie in non-invariant components of G.

D.17. Proposition. Let (G, d) be a quasifuchsian group. Then G has no accidentalparabolic transformations.

Proof. Since G has two invariant components, call them .J and d', the entire limitset is the common boundary of these two components. Suppose j e G is accidentalparabolic. Then the axis A of j lies, except for the fixed point x of j, entirely in A.Hence d' lies entirely in one of the open discs bounded by A; call it B. Let B' bethe other open disc bounded by A. Since d' c B, d' c B. Since A intersectsthe limit set only at x, every limit point other than x lies in B. We conclude thatB' contains no limit points of G. By D.15, j is not accidental.

D.18. Let (G, A) be a non-elementary B-group. The number of 00-connectors T2(this terminology will be explained in X.D) is defined to be the number of non-


conjugate maximal cyclic subgroups of accidental parabolic transformationsin G.

D.19. Proposition. Let (G, A) be a B-group, where A/G has signature (p, n; v1,..., v.).

Then

(i)r253p-3+n.(ii) Let J1, ..., Jk be a complete set of maximal non-conjugate cyclic subgroups

of accidental parabolic transformations. Let A;,, be the axis of If A', is the trueaxis of Jm, set A. = Otherwise, there is a half-turn jm in the stabilizer of thefixed point of Jm, set A. = A, U jm(A,). Then (A . , ... , Ak) is precisely invariantunder (Jl,...,Jk) in G.

Proof. Statement (ii) follows almost at once from D. 14, together with the fact thatif A. is not the true axis, then Am is precisely invariant under the stabilizer ofthe fixed point of Jm. Then the axes (or pairs of axes) A, , ... , A,, project to disjointsimple closed curves on A/G. Then by V.G.15, there are at most 3p - 3 + n ofthem.

D.20. The proposition above states that given the non-elementary B-group (G, A),the accidental parabolic transformations in G can be recognized by a system ofhyperbolic loops on A/G, where each loop in the system determines a primitiveaccidental parabolic transformation (see V.G.15).

D.21. Theorem. Let (G, A) be a non-elementary B-group. Then either G is quasi-fuchsian, or G is degenerate, or G contains accidental parabolic transformations.

Proof. If A is the only component of G, then G is degenerate. Assume there isanother component A', and let H = Stab(A'); then H has two invariant compo-nents Ao A and A'. If G = H, then G is quasifuchsian. From here on we assumethat G 0 H; i.e., we assume that H is neither degenerate, nor quasifuchsian. Wealso assume that G contains no accidental parabolic transformations.

Let g e G - H, then g(A') is different from both Ao and A'; since G has at leastthree components, G has infinitely many components (V.E.9).

Let fl: A - H2 be a Riemann map. Just as we defined axes of elements of G,so we can define the convex region for any subgroup J of G; let be theconvex region in H2 of f,(J), then the convex region K(J) = f-'(K(f*(J))) (seeV.G.13).

For some fixed element g of G - H, consider the sets K(H), and K(gHg-')in A; assume that the intersection of these sets has non-empty interior U. We useV.E.5 (see also V.I.18) in H2 to observe that for every point in the tangent spaceof fl(U), there is a nearby point that coincides with an axis of a hyperbolic elementof H; similarly with gHg-'. Hence there is a point z e U which lies on the axis Aof a e H, and lies on the axis B of b e gHg -'. Since G has no accidental parabolictransformations, a and b are both loxodromic. The endpoints of A are the fixed

226 IX. B-Groups

points of a on o4'; hence there is also an axis A' in d' for a. Similarly, there isalso an axis B' for b in g(A'). The axes A and A', together with the fixed pointsof a, make up a simple closed curve; so do B and B', together with the fixed pointsof b. These two simple closed curves meet at z, and can meet at no other point,for d' and g(A') are disjoint, and two distinct axes in A meet in at most one point.We have shown that for every g e G - H, K(H) and K(gHg-' )) = g(K(H)) havedisjoint interiors; i.e., int(K(H)) is precisely invariant under H.

Since H is non-elementary, int(K(H)) 96 0. Since int(K(H)) is precisely in-variant under H in G, and [G: H] > 1, H is of the second kind (for a Fuchsiangroup of the first kind, the convex region is the whole disc). Further, the hyper-bolic area of K(H)/H is less than the area of A/G (the statement that int(K(H))is precisely invariant under H in G is equivalent to the statement that there isan isometric embedding of int(K(H))/H into A/G; the metric in both cases isof course the Poincare metric). Hence f,(H) is a Fuchsian group whose convexregion, modulo the group, has finite area; i.e., f,(H) is finitely generated (seeV.G.l4 and the references in V.J).

The smallest normal subgroup of G containing H has at least three compo-nents: Ao, A', and g(A' ); hence it has infinitely many components. It follows that[G: H] = oo; hence the area of HZ/#,(H) is infinite. It follows that f,(H) isa geometrically finite Fuchsian group of the second kind. Then (VI.F.4) thestabilizer of every boundary axis is hyperbolic.

Let A be a boundary axis of K(H) in A, and let j be an element of StabG(A).Since G contains no accidental parabolic transformations, j is loxodromic. Thenj has an axis in A and an axis in A'; these two axes can be combined to forma simple closed curve W. Let B be the topological disc bounded by W, whereB fl A is precisely invariant under Stab(A) in H. Then for every h e H - Stab(A),h(A) fl B = 0, hence h(B) fl B = 0. We conclude that B is precisely invariantunder Stab(A) in H. Hence there are no limit points of H, other than the fixedpoints of Stab(A), in B. This contradicts the fact that there are limit points ofH in B separating Ao from A'. We have shown that j is parabolic, contradictingthe assumption that G contains no accidental parabolic transformations. Q

D.22. We remark that the three possibilities mentioned in D.21 are not mutuallyexclusive. There are degenerate groups with accidental parabolic transformations(see 1.11).

IX.E. An Isomorphism Theorem

E.I. One of the most important theorems in the theory of Fuchsian groups is theNielsen isomorphism theorem. This theorem asserts that if a type-preservingisomorphism between Fuchsian groups preserves boundary elements in bothdirections, then it is geometric; that is, such an isomorphism is induced by a

IX.E. An Isomorphism Theorem 227

homeomorphism of H2 onto itself, and, except perhaps for orientation, thishomeomorphism is the restriction of a deformation. A proof of this theorem isbeyond the scope of this book, but we will prove a weaker version; our additionalhypotheses are that the isomorphism is type-preserving, and that it preservesboth intersection and separation of axes in both directions. Also our conclusionhere is only the existence of a homeomorphism from H 2 onto itself (but see F.9).The proof uses building blocks; that is, we will show that the theorem is true forcertain two generator Fuchsian groups, and then use combination theorems toglue together the homeomorphisms.

For each Fuchsian group F containing half-turns, there are certain elementsfor which we make a choice of axis, as opposed to true axis. It is easy to see thattwo axes intersect if and only if the corresponding true axes intersect; similarly,except for coincidence, an axis separates two others if and only if the corre-sponding true axis separates the corresponding true axes.

E.2. Let F and F be Fuchsian groups. An isomorphism cp: F -. F, is calledhyperbolic if p is both type-preserving and preserves intersections and separa-tions of axes. That is, for every pair of hyperbolic elements f and g of F, the axisoff in 0--02 crosses the axis of g if and only if the axis of (p(f) crosses the axis ofV (g); also, for every triple f, g, and j of hyperbolic elements of F, with disjointaxes, the axis of j separates the axes of f and g, if and only if the axis of V(j)separates the axes of p(f) and (p(g).

The isomorphism gyp: F -+,P is called geometric if there is a homeomorphism4: fH02 - H2 inducing V.

Theorem (Nielsen [78]). Let F be a finitely generated Fuchsian group, and lettp: F - P be a hyperbolic isomorphism onto another Fuchsian group. Then 0 isgeometric.

E.3. We first take up the case that F is elementary. Then F is either trivial, ellipticcyclic, parabolic cyclic, hyperbolic cyclic, or 712:712. If F is trivial, there isnothing to prove. If F is elliptic or parabolic cyclic, then let a be (geometric)generator of F. Since p(a) is a (geometric) generator for P, there is a a/i c- RAJwith qi o a o 0-' = P(a)

If F is hyperbolic cyclic, normalize so that a(z) = Az generates F; also nor-malize P so that it is generated by cp(a)(z) = a(z) _ .Iz. Set a = log(A)/log(A), setcli(reie) = rete, and note that qioa = cp(a)ogi.

If F = 712 * 712, then normalize F so that it is generated by a(z) = 11z, andb(z) = Az. Similarly normalize P so that (p(a) = a, and cp(b)(z) = b(z) = Jz. Thesame function t/i(re'e) = r°e'° commutes with a, and conjugates b into bb.

From here on, we assume that F is non-elementary.

E.4. Lemma. If a is a (hyperbolic) boundary element of F, then (p(a) is a boundaryelement of F.

228 IX. B-Groups

Proof. Let H be the boundary halfplane bounded by A, the axis of a. Since P isnon-elementary, one of the halfplanes bounded by A contains axes of hyper-bolic elements, call the other one P. Since (p and gyp-' preserve intersection andseparation of axes, no axis of P lies in A`, and no axis of P crosses X. Then forf E P. the only way we can have f($) fl P # 0 is for f(A) = I. This shows that13 is precisely invariant under Stab(A' ). 0

E.S. We first prove our theorem in the special case that °S = (°Q(F) fl H2)/F hassignature (0, 3). That is, we assume that either S = H2/F is a sphere with threespecial points, or a disc with two special points, or an annulus with one specialpoint, or a sphere with three disjoint closed discs removed (a sphere with threedisjoint closed discs removed is sometimes called a "pair of pants").

In all these cases, we can find a set of generators a, b, c for RI (°S), where eachof these generators is defined by a simple loop from the base point that windsonce around a boundary component of °S; the three loops are essentially disjoint(that is, they are disjoint except at the base point, and they do not cross at thebase point); and the product c b a = 1. Using the natural homomorphism, wecan find corresponding generators for F, which we also call a, b, and c.

We define the axis A to be the ordinary axis of a if a is hyperbolic, the fixedpoint of a in l l2 if a is elliptic, or the fixed point of a if a is parabolic. We likewisedefine the axis B of b, and C of c.

For each pair of the these generators, there is a unique hyperbolic line, orline segment L' orthogonal to both axes. For example, in the case that say a isparabolic, and b is hyperbolic, L' is a half line with its infinite endpoint at thefixed point A of a, and its finite endpoint on the axis B of b; of course, L' isorthogonal to B. Let L be the full hyperbolic line on which L' lies.

We are free to cyclically permute the elements a, b, c. For reasons that willbecome apparent in the next section, we require that if any of these three elementsare hyperbolic, then c is.

We can write a as a product of two reflections; first reflect in some line M,and then reflect in L. Similarly, we can write b as a product of first reflect in L,and then reflect in some line N. Note that if a is hyperbolic, then L is orthogonalto A; if a is parabolic, then L ends at A, and if a is elliptic, then L passes throughA. Similarly, M is orthogonal to, or passes through or ends at B. Of course, c-'is then the composition of: first reflect in M, then reflect in N.

Let M' = a(M), and let N' = b-' (N). It might be (although in fact it does nothappen) that c is hyperbolic and M and N lie on opposite sides of L; in this case,simply replace b by b-'. If M and N intersect inside H2, then the point ofintersection is of necessity a fixed point of c; in this case, c is elliptic. Similarly,if M and N meet at a point on the circle at infinity, then that point is a fixedpoint of c, and c is necessarily parabolic. In the remaining case, the commonperpendicular of M and N is the axis C of the hyperbolic transformation c(see V.B for the analogous discussion in H'; replace half-turns by reflections, andthe arguments become equally valid in H2).


Fig. IX.E.1

The lines M, M', N, and N' bound a polygon D; the sides of D are identifiedby the elements a and b; it is easy to verify that the hypotheses of Poincare'spolygon theorem are satisfied (the only thing remaining to verify is the conditionon the sum of the angles if M and N intersect; this follows at once from the factthat D is symmetric about L, and that c is the composition of reflection in Nfollowed by reflection in M).

E.6. We next consider the group F generated by a = cp(a) and b = Wp(b). Exactlyas above, we define the axes of a and b, construct the line L, joining, or orthogonalto these axes, and construct the lines ai, R, M', and R'.

We need to observe that if c is hyperbolic, then a and R lie on the same sideof L. If this were not the case, then the axis C of c' would cross L, separating d,the axis of a, from D. Since c is a boundary element, so is c. Hence one of thehalf-planes bounded by C has no limit points on its boundary and contains noelliptic fixed points. It follows that C cannot separate d from D.

Exactly as above, we can construct the fundamental polygon D, where again,the intersection of the sides 1N and R (and the intersection of the sides a' andR') is completely determined by the type of c. If c is elliptic, then the angle ofintersection is also determined. We conclude that there is a homeomorphism 0from the closure of D onto the closure of b that commutes with the identifica-tions; that is, on M, , o a = d o 0, and on N', o b = b o 0. There is no difficultyin using the action of F and F to extend /i to a homeomorphism of H2 whichrealizes gyp.

We make one further requirement of this homeomorphism 0; we requirethat 0(A, B, C) = (d , B, C). If for example a is not hyperbolic, then it is norestriction to require that ¢(A) = T. It is easy to choose i/i1D so that it maps(A fl D, B fl D, c fl b) onto the corresponding parts of D. Since A, B and C are allorthogonal to the sides of D, i/i(A, B, C) = ( 9 ,0 ,C ).

E.7. We turn now to the general case, where F is a finitely generated Fuchsiangroup; set °S = (H2 fl °12(F))/F. Then °S is a finite marked Riemann surface; letw1, ..., wk be a maximal system of loops on °S and let Y,, ..., Y. be the buildingblocks, or units, of the system (see V.G. 15).

230 IX. B-Groups

We assume we are given a type-preserving isomorphism rp from F ontoanother Fuchsian group F, where two hyperbolic elements of F have intersectingaxes if and only if their images under (p have intersecting axes, and the axis off e F separates the axes of g and h if and only if the axis of qp(f) separates theaxes of (p(g) and cp(h).

Let Y = Y. for some m, let T c LO2 be some connected component of thepreimage of Y, and let G = Stab,(T). One sees at once that Ii2/G is a planarsurface with three boundary components. Let a, b, and c, be the three generatorsof G; note that if any of these is hyperbolic, then it is a boundary element of G.Since cp preserves intersections and separations of axes, C = rp(G) is generatedby three elements, whose product is the identity, and if any of these three genera-tors is hyperbolic, then it is a boundary element. We conclude from the analysisin E.5-6, that for each Ym, and for each choice of Tm, lying over Ym, there isa homeomorphism 0 that conjugates G. = Stab(T,) onto (p(Gm).

E.B. We next use combination theorems to glue together these different homeo-morphisms. Start with some Y, , choose some T, lying over Y, , let G, = Stab(T, ),and let 0,: 0-fl2 - H2 be the homeomorphism which conjugates G, onto 6, =qp(G, ). Choose Y, so that Y, and Y2 have at least one common boundary curve,and choose T2 so that T, and T2 have a common geodesic, call it W, on theirboundary. Set G2 = Stab(T2). Looking from the point of view of G,, W is aboundary geodesic, and the boundary half-plane is precisely invariant under thehyperbolic cyclic group J = Stab(W). Looking from the point of view of G2,W is also a boundary axis, but now the other half-plane is precisely invariantunder J. Extend W to a full circle, and observe that this circle bounds two closeddiscs B, and B2, where B,, is a strong (J, Gm}block (if W is the true axis, then B.is precisely invariant under J in Gm). The homeomorphisms 41, and 02 both mapW onto the same axis IV, and, again using the fact that cp preserves separationand intersection of axes, ry,(T1) and fi2(T2) are disjoint. A

Suppose (13, , 132) is not a proper interactive pair. Then every point of 2 is aG,-translate of some point of &,; this implies that G, is elementary. Then thereis half-turn g, a G, with g, (W) a B2. Similarly, G2 is elementary, and there isa half-turn g2 e G2, with g2(W) c B1. This yields three distinct axes for J whichcannot be. We conclude that (C 132) is proper.

We now use VII.C.2(viii) to conclude that the convex region of <G,, G2),factored by <G,, G2) is the union of T,/G, and T2/G2, with the boundarycomponents corresponding to W identified.

The two homeomorphisms, 01 and 1'/2, when restricted to W, are bothhomeomorphisms of W onto the same axis, and both have the same invarianceproperty; in particular, 0,(W) and 4i2(W) have the same orientation. It is easyto modify them in a neighborhood of W, so that they agree on W. We now definea new map 4P2 by 012 IT, = 41,, and ,12I T2 = 412, and then we extend 4112 asfar as possible by the requirement that 41120g(Z) = cp(g) u 12(z) forge <GG2).Since T. is precisely invariant under G. in all of F, we know that *12 is well


defined, is locally a homeomorphism, and satisfies the right invariance properties.However we do not immediately know that 4/1' is injective.

Each of the regions T. lies over some Y., is hyperbolically convex, and isbounded by axes. Wherever defined, 412 maps each such region onto a regionbounded by disjoint axes, hence hyperbolically convex. Since cp preserves separa-tion and intersection of axes, it must be that distinct such regions are mappedby 0 onto distinct such regions; of course, two adjacent regions are mappedonto adjacent regions. We conclude that 0" is injective. (This argument needsmodification in the case that one or more of the boundary axes of the T. are nottrue axes. If this occurs, it is easy to extend 0, and 02 so that they map true axesonto true axes, and so that they map the regions bounded by true axes (these areslightly larger than the regions bounded by the corresponding axes) onto regionsbounded by true axes.)

E.9. We continue in the above manner until we have a homeomorphism 0'defined on a full set of preimages of the Y.. Of course ii' is defined on a connectedset of these preimages; call it T If T is not all of 112, then since T is boundedby axes of hyperbolic elements of F, there must be two such axes that areF-equivalent, but not equivalent under Stab(T). Call these axes W, and W2, LetJ. = Stab(Wm), and let f be an element of F mapping W, onto W2. We adjoin fto Stab(T), to get a new subgroup H of F, and we get a new region T', where T'is the union of the H-translates of T There are closed outside discs B, and B2,bounded by W, and W2, respectively, so that f maps the outside of B, ontothe inside of B2. It is clear that there are points of H2/F not in the projection ofB, U B2. Then by VII.E.5, H = Stab(T)* f, and the convex region of H, factoredby H, is T/F with the two boundary components corresponding to W, and W2glued together.

We want to extend O' continuously to T'. It is clear that we can do this oncewe have the new /i' continuous near both W, and W2. In order to achieve this,we deform O' near W2 so that on W1, i/r' o f = (p(f) o 0', and so that 41' still mapsW2 onto the corresponding axis in F. We observe exactly as above that the new0' is a homeomorphism of T' onto a region bounded by certain axes; if W. is anaxis on the boundary of T', then the axis of V(W.) lies on the boundary of i/i'(T').

E.10. The total number of boundary geodesics for any of the subgroups mentionedabove is at most 2k, where k is the number of geodesics in the system of loops.Each time we perform the operation described in E.9, we decrease the numberof boundary geodesics by two. Hence after a finite number of steps, we arriveat a subgroup F0 of F where F0 and F agree in their actions on each of theregions T, and they have exactly the same boundary elements; then, necessarily,F0=F. O

E.1 1. We will need the following special case of E.2.

232 IX. B-Groups

Corollary. Let cp: F - F be a hyperbolic isomorphism, where F is a finitelygenerated Fuchsian group of the first kind, and F is Fuchsian. Then P is alsoof the first kind, and there is a (possibly orientation reversing) type-preservingsimilarity between F and F.

Proof. It suffices to prove that F is of the first kind, for then the result is just arestatement of E.2. If F were of the second kind, then it would.have a boundaryaxis, and then by E.4, so would F. El

IX.F. Quasifuchsian Groups

F.I. Recall that G is a quasifuchsian group (of the first kind) if G is a Kleiniangroup with two invariant components, d and A', where A/G is a finite Riemannsurface. Fuchsian groups of the first kind are special cases of quasifuchsiangroups; others were constructed in VIII.E.3. We mention Bers' simultaneousuniformization theorem, which asserts that given any two finite marked hyper-bolic Riemann surfaces and an orientation reversing homeomorphism betweenthem, there is a quasifuchsian group G, so that Q(G)/G is (conformally equivalentto the union of) these two surfaces; also, there is a homeomorphism from A ontod' that covers the given homeomorphism between the surfaces and commuteswith every element of G.

F.2. Let G be a quasifuchsian group with components A and A', and let j be aloxodromic element of G. Let fi, #' be the Riemann map from respectively,onto the upper and lower half planes, respectively. Let l'', W be be the axis off;(j), respectively, and let W = #`(W), W' = (if the axes are not thetrue axes, then there is some choice involved; see F.3). The curves W and W' arecalled, respectively, the axes of j in A, A'.

Lemma. Let W be the axis of j in A. Any sequence of points of W accumulatesonly to a point of W or to a fixed point of j.

Proof. Let {x,, } be a sequence of points of W, where x. - x e C, and let D bea fundamental set for J = Q>, where D is bounded away from the fixed pointsof j; for example, choose D to be an annulus, including one boundary curve.For each m, find jm E J, so that fm(m)E1) . Note that for m - +oo, jm(D)converges to the attracting fixed point of j, and for m - -oo, jm(D) converges tothe repelling fixed point of j. Hence x is either a point of W, or one of these twopoints. p

F.3. The full axis W of j is the union of W, W', and the two fixed points of j; thecontent of the above lemma is that W is a simple closed curve. In the case that

IX.F. Quasifuschsian Groups 233

W is not the true axis, then there are two choices each for W and W'. Choosethese so that if h is a half-turn stabilizing the true axis, then h(W) and W touchbut do not intersect at the fixed points of J.

Two translates g(W) and h(W) are distinct if and only if g and h lie in differentleft cosets of G/<j>. Of course, distinct translates of W need not be disjoint.

If the primitive loxodromic element j e G represents a simple loop in A, thenits axis W is precisely invariant under its stabilizer. We have defined the axis, asopposed to the true axis, so that Stab(W) is loxodromic cyclic.

Lemma. Let G be a quasifuchsian group, let J he a maximal cyclic loxodromicsubgroup of G, where J represents a simple loop in A (that is, the axis of J in A isprecisely invariant under its stabilizer) and let W be the full axis of J. Then Wis a strong (J, G)-block.

Proof. Let g be some element of G - J. Since W projects to a simple loop, notranslate of W can cross W. Two axes in the lower half-plane can intersect inat most one point, so if g(W) fl W # 0, then these two loops meet at one of thefixed points of J. Since G is discrete, this can happen only if they meet at bothfixed points of J; we have shown that w fl g(W) # 0 if and only if g stabilizesthe pair of fixed points of J. Since go J, this can occur only if W is not the trueaxis, and g is a half-turn stabilizing the true axis. In this case, there is only theone translate of W intersecting W, and the points of intersection are exactly thefixed points of J. 0F.4. Lemma. Let J be a loxodromic cyclic subgroup of the quasifuchsian group G.Let W be a simple J-invariant loop, where W fl A(G) = A(J). Let {g.(W)} be asequence of distinct translates of W. Then dia(g.(W)) - 0.

Proof. The proof is an easy adaptation of VII.B.9-14. Normalize G so thatoo a °Q(G), and so that J is generated by j(z) = Az, IA.l > 1. Let D = {z11 5 Iz1 SIAl } be a fundamental set for J. Observe that w fl D is bounded away from A(G).For each m, replace gm by an element of the form g. o jm, j, a J, so that the centerof the isometric circle of the new g. lies in D. These centers accumulate onlyat A(G), and the radii tend to zero. Hence for m sufficiently large, W lies outsidethe isometric circle of g.. Then gm(W) lies inside the isometric circle of g,,', andthe radius of this circle tends to zero. 0

F.S. Let (F, A) and (P, d) be quasifuchsian groups. As in the Fuchsian case, anisomorphism q : F - F, is called hyperbolic if cp preserves intersections andseparations of axes.

Note that a Riemann map fl: A - Ol2 induces a hyperbolic isomorphism; infact, any type-preserving similarity A --+.3 induces a hyperbolic isomorphismbetween F and F. Note also that by D.17, the isomorphism induced by P istype-preserving.

234 IX. B-Groups

Using F.3, it is easy to see that if G is a quasifuchsian group with componentsd and A', then the identity isomorphism between (G, A), and (G, A') is hyperbolic.Of course the identity isomorphism is type-preserving.

F.6. Theorem. Let (G, A) be a quasifuchsian group, let fl: d - H2 be a Riemannmap, and let F = f,(G) be the Fuchsian model of (G, A). Then f-' is the restrictionof a deformation; that is, there is a homeomorphism q': C -' C, with of=(f-'),(f)o4, for all feF.

Proof. The triangle groups were taken care of in IX.C, so we can assume thatG is not a triangle group.

Let cp be the isomorphism Let f' be a Riemann map from A' ontothe lower half-plane L, and let (F', L) be the corresponding Fuchsian model. Theisomorphism ip' = (fl'), o (p can be thought of as the composition of first ip, taking(F, H 2) to (G, A), then the identity, taking (G, A) to (G, A'), then (fl'),, taking (G, A')to (F', L). Since all three of these isomorphisms are type-preserving and hyperbolic,so is the composition gyp'. Then by E.11, there is a type-preserving but possiblyorientation reversing similarity ii' from (F, L) onto (F', L) that induces rp'. Wenow define the homeomorphism qi: Q(F) -> Q(G) as follows. For ze H2, 41 (z) =fl`(z), and for z e L, O(z) = (fl')-' o 0'(z). It is easy to check that iji o f(z) =rp(f) o ii (z) for all f e F, and for all z c- Q(F).

If x is a limit point of F, where x is not a parabolic fixed point, then (V.G.17)there is a simple axis W, and there is a sequence { fm} of elements of F so thatfm(W) nests about x. Let W be the corresponding full axis in G, and let gm = cp(fm).Then the axes gm(l) have the right separation property, and since W is a G-block,dia(gm(W)) - 0. Hence (gm(W) nests about some point, call it 4i(x).

Observe that i/i, as we have defined it, is continuous from inside Q(G) at thenon-parabolic limit points.

F.7. If x is a parabolic fixed point of F, then define fi(x) to be the fixed point ofthe corresponding parabolic element of G. It is immediate that /i is continuousat x from inside any horoball centered at x.

If {xm} is a sequence of points approaching x from inside H2, but outsidesome fixed horoball B centered at x, then we can find elements jm a Stab(x)so that ym =j-'(x.),is bounded away from x. If the points y, all lie in acompact part of H', then their image under 0 lies in a compact part of Q(G), so(p(jm) o 4,(ym) = 0 ojm(ym) = fr(xm) necessarily tends to O(x). If the points ym tendto a point y on i)H2, necessarily different from x, then there is some axis A of Fthat separates all the ym from 8B. Since A does not pass through x, jm(A) - x.Complete fr(A) to a loop by adjoining the fixed points of the (p-images of Stab(A).The loop fi(A) bounds two discs; call the disc containing the parabolic fixed pointfi(x) the outside. Consider , o jm(A) = o VI(A). By F.4, the diameter of thissequence of loops tends to zero. Since the points fi o jm(ym) = fr(xm) all lie inside1'(A), O(xm) - x.

IX.F. Quasifuschsian Groups 235

F.B. It is immediate from the way we have defined qi that it is an injection fromC into itself, and that the image of C covers Q(G). If y is any limit point of G,then there is a sequence of points of 0, call it {kxm)}, approaching y. Choosea subsequence so that x, - x. Note that x is necessarily a limit point of F; hencei/i(x) = y. Since i/i is bijective, continuous and conformal on H2, it is an orientation-preserving homeomorphism.

F.9. In the case that G is Fuchsian, F.6 gives us a stronger version of E.2.

Theorem (Nielsen [78]). Let cp: F -+ G be a type-preserving hyperbolic isomorphismbetween two finitely generated Fuchsian groups of the first kind. Then there isa possibly orientation reversing deformation 41 of F onto G inducing cp.

F.10. We can now sort out the two seemingly different definitions of "quasi-fuchsian group" (see VIII.E.3 and B.2).

Corollary. Every quasifuchsian group is analytically finite.

Proof. Let (G, d) be a quasifuchsian group, let F be the Fuchsian model of G, andlet d' be the other component. By F.6 there is a deformation 0 of F onto G, sothat d'/G is homeomorphic to L/F. Hence d'/G has finite genus, and has at mosta finite number of boundary components. Each boundary component of d'/Gcorresponds to a maximal cyclic subgroup <g>. Since F is of the first kind, thecorresponding element f e F is necessarily parabolic, and there is a preciselyinvariant horoball. Then g is parabolic, and the image under i/r of this horoballis a precisely invariant topological horoball near the fixed point of g. Normalizeso that the fixed point of g is at oo. Since any topological precisely invarianthoroball contains a circular precisely invariant horoball, the boundary com-ponent of d'/G is a puncture.

F.11. The reflection z - z induces an isomorphism of a Fuchsian group F ontoitself that commutes with every element of F. Since every quasifuchsian group isa deformation of a Fuchsian group, for every quasifuchsian group G, there isa homeomorphism 0 that interchanges the two invariant components, andcommutes with every element of G. Since the map z - z reverses orientation, sodoes the map >/i.

F.12. Theorem. Let (G, d) and (ti, d) be quasifuchsian groups with the same basicsignature. Then there is a deformation i/r of G onto 0, where #(d) = 3. Further,if we are given a type-preserving similarity 0o: A - d, then >ji can be chosen so that1k, = ('o)*

Proof. Since G and ti have the same basic signature, we can assume that we aregiven kio. Let (F, H2) be the Fuchsian model for (G, d), and let (F, H2) be theFuchsian model for (0, d). By F.6, there is a deformation 0 1 from F onto G, and

236 IX. B-Groups

there is a deformation 02 from F onto C. Also, 02' o o o 0, is a type-preservingsimilarity of F onto F. By F.9, there is a deformation 03 from F onto F, where

' is a deformation of(413). = [4, ' o 4,0 o,/i, ],,. Then the composition 02 o 03 o 411G onto C whose induced isomorphism is qio.

IX.G. Degenerate Groups

G.I. Recall that a B-group is degenerate if it has exactly one component, neces-sarily simply connected (one also sometimes speaks of any Kleinian group, whichis finitely generated but not geometrically finite, as being degenerate). Afterproving the existence of degenerate groups, we will show that degenerate groupsare finitely generated but not geometrically finite.

G.2. We know from C. I that a degenerate group cannot have signature (0, 3;V1, v2, v3); that is, its Fuchsian model cannot be a triangle group. We will needthe following fact about Fuchsian groups.

Lemma. Let F be a finitely generated Fuchsian group of the first kind, withsignature a # (0, 3; v,, v2, v3). Let w he a simple loop on °S = H2 fl °Q(F))/F, wherew is either non-dividing, or w divides °S into two subsurfaces, each of which eitherhas positive genus, or has at least three boundary components. Then w is hyperbolic.

Proof: The hypotheses on w are equivalent to saying that w is a simple loop thatdoes not bound either a disc or a punctured disc on °S. It follows from F.12 thatthe problem is purely topological; that is, if there is one Fuchsian group ofsignature a on which w is hyperbolic, then the corresponding loop w is hyperbolicon every Fuchsian group of signature a. The construction of one such group isan exercise (see VIII.H.6).

G.3. Theorem. Let a = (p, n; v,_., be the signature of a non-elementary Fuchsiangroup, where a (0, 3; V1, V2, V3)- Then there is a degenerate group G, so that Q/Ghas signature a.

Proof. Let F be a Fuchsian group of signature a; let w be a hyperbolic simpleloop on °S as in G.2, let j be a hyperbolic element of F determined by a lift of w;and let A be the axis of J. We choose A to be the axis, rather than true axis, sothat J = Q> is the stabilizer of A. There are two cases to consider, according asw is dividing or non-dividing; we first take up the case that w is dividing.

Let £ be the set of translates of A under F. The axes in £ divide H2 intoregions; let E, and E2 be the regions on either side of A, and let F. = Stab(Em).In the special case that the signature of F is (0,4;2,2,2, v), one of these regions isbounded by exactly two axes, and is precisely invariant under an infinite dihedral

IX.G. Degenerate Groups 237

group containing J. in this case, call that region E,. In any case, since F isnon-elementary, we can require that F2 be non-elementary.

We remark that since w is dividing, E, and E2 are inequivalent under F. Itis almost immediate from the way E. and F. have been defined that Em isprecisely invariant under F. in F.

We next replace A by its full axis A; and also replace all the correspondingtranslates of A by their full axes. When we do this, we also replace Em by theunion of E. with its reflection in the real axis, together with the interior points,if airy, of the common boundary of these two sets. While the sets E, and E2 havebeen enlarged, the subgroups F, and F2 remain unchanged.

The complete axis A divides C into two closed discs, B, containing E2, andB2 containing E,. Every element of F, either keeps A invariant, or it maps Aonto an axis on the boundary of E,; hence 6, is precisely invariant under J inF,. If A is also the true axis, then in fact the closed disc B, is precisely invariantunder J in F,; if A is not the true axis, then F, is elementary, and B, is notprecisely invariant, but is a (J, F, )-block. Similarly, A2 is precisely invariantunder J in F2. Even if A is not the true axis, none of the half-turns with fixedpoints on the true axis lie in F2, so, in any case, B2 is precisely invariantunder J in F2. Since F2 is non-elementary, there are points of H2 that are notF2-equivalent to any point of B2; hence is a proper interactive pair.We have shown that the hypotheses of VI1.C.2 are satisfied for the groupF' = <F1,F2>.

G.4. Proposition. F' = F.

Proof. Let g be some element of F. Pick a point z on A and connect z to g(z) bya (hyperbolic) line C. Since J = StabG(A), J c F', so we can assume that g(z) isnot on A. Suppose that C passes through E,; the argument is essentially the samein the other case. After passing through E,, C leaves by crossing some translateA, = f, (A) on ME,. Since w is dividing, the intersection of C with A and the inter-section of C with A, have the same orientation; i.e., the element f, mapping Aonto A, preserves E,. We conclude that f, e F and that if C does not end atA then C continues on past A, into f,(A2). Then there is an element f2 C if, FZf1 'mapping A, onto A2, the translate of A crossed by C as it leaves f,(A2), and soon. We have shown that F' = <F, , F2> = F. Q

G.5. We observed above that E. is precisely invariant under F. in F. It followsthat Em/Fm is embedded in ai12/F. In fact, by VII.C.2(viii) H2/F is the union ofK(F1)/F, and K(F2)/F2, where these two surfaces are joined along the commonboundary loop w (K(F) is the convex hull of the limit set of F). Hence the(hyperbolic) area of K(Fm)/Fm is not greater than the area of 0-12/F. Since the areaof 0I2/F is finite, so is the area of K(Fm)/Fm, from which it follows that F. is finitelygenerated.

238 IX. B-Groups

G.6. Now that we have constructed F, we can slide and bend the constructionas in VIII.E.2-3. Normalize F so that J has its fixed points at 0 and oo, and sothat F acts on 112. Let t e C, and let k,(z) = tz. Let G(t) = <F,, F2(t)>, whereF2(t) = k,F2k('.

For each t there is the natural homomorphism cp,: F - G(t), defined as theidentity on F,, and as the obvious isomorphism, f -a k, of o k-', on F2.

Lemma. For every f e F, the fixed points, and the square of the trace, of cp,(f) areholomorphic functions of t.

Proof. Pick generators for F, and F2. Then tr2(f) is a polynomial in the entriesin these generators. The images of the generators for F, in G(t) are identical withthe generators in F,, and the entries in the images of the generators of F2 in G(t)are polynomials in the following variables: the entries in the generators of F2,/t and 1/ f . Similarly, the fixed points of q,(f) are solutions of a quadraticequation whose coefficients are polynomials in these variables.

G.7. For each hyperbolic f e F, there are only countably many complex numberst so that tp,(f) is parabolic. -Since F is countable, there are only countably manyp for which there is a t = pe'B so that for some hyperbolic f e F, (p,(f) is parabolic.Fix p = po so that tr2((p,(f )) 96 4 for all hyperbolic f e F, and for all t = poe'B.

For t = po, G(t) preserves 02, and, by VII.C.2(ii) it is discrete; i.e., it isFuchsian. It is easy to see, using the same argument, that for t = poe'a, 0 small,G(t) is quasifuchsian (see VIII.E.3).

Let T be the set of numbers t = poeie for which there is a loop W(t) dividingC into two closed discs, B, (t) and B2(t), where B,(t) is a (J, F, )-block, and B2(t)is precisely invariant under J in F2(t). For t c- T, the groups F, and F2(t) satisfythe hypotheses of VII.C.2; it easily follows from this theorem that for each to T,G(t) is quasifuchsian. By F.12, there is a homeomorphism ,: Q(F) -+ Q(G(t)),where 0, induces the type-preserving isomorphism rp,: F - G(t). Since (p,IJ = 1,0/i, maps A onto a J-invariant loop; it is easy to deform 4i, near A so thatOJA) = W(t).

G.S. We now fix the choice of F, and F2, so that in the original Fuchsian groupF, B, is the left half-plane, and B2 is the right half-plane.

For every t E T, the limit set A (G(t)) is a simple closed curve passing through0 and cc; the complement, Q(G(t)) has two components, the upper component4(t), and the lower component 4'(t).

Proposition. If t e T, and Im(t) > 0, then 4(t) 012.

Proof. The limit set A, of F, lies on the positive reals, R,, including 0 and oo.The limit set A2 of F2(t) lies on a ray R2 in the third quadrant, again including0 and cc (see fig. VIII.E.8). For any f, e F f, (R2) is contained in the lower half


plane. Likewise, for any f2 e F2(t), f2(R,) is contained in the smaller sectorbetween R, and R2. Then f, o f2(R,) is also contained in that smaller sector,between R, and ft(R2); likewise, f2 o fl(R2) is contained between f2(RI) and R1.Continuing in this fashion, we see that all the translates of both A (F,) and A (F2(t))are contained in the closed sector between R, and R2 in the lower half-plane.Since the limit set of G(t) is the closure of the translates of A, and A2, A(G(t))does not intersect 112.

G.9. It is clear that we can interchange the roles of F, and F2(t) in the aboveargument. This shows that A(t) also contains the half-plane {arg(t) < arg(z) <arg(t) + tt).

G.10. Lemma. T is an open subset of the circle of radius po.

Proof. Let t c- T, and let W(t), bounding the discs B, (t) and B2(t), be the simpleclosed curve given by the definition of T. Let E be a fundamental set for J,where E is bounded away from both 0 and oo. Then there are only finitely manyG2(t)-translates of B2 intersecting E in a sufficiently small neighborhood of W.Hence it suffices to choose t' so close to t so that after rotating these finitely manytranslates of B2(t) by arg(t - t') they are still disjoint from W(t) = W(t'). Theother G2(t') translates of W(t) that intersect E remain far from W(t). Of course,there are translates of W(t) other than those intersecting E; that these others donot intersect W(t) follows from the J-invariance of W(t).

G.11. The point to = po is surely in T. We start at this point, that is, at arg(t) = 0,and traverse counterclockwise until we reach some first point t* not in T. Thatthere is such a point follows from the fact that at arg(t) = it, the elements of G(t)all preserve H 2; if - po were in T, then the limit set of G(- po) would be containedin the right half-plane, so G(t) would not have two components. Set G = G(t*).

G.12. Let To be the arc of T between to and t *. As t traverses To counterclockwisefrom to to t*, in some sense the upper component A(t) gets larger, and the lowercomponent A'(t) gets smaller. We now fix a fundamental polygon P, for F, in012, and likewise fix a fundamental polygon P2 for F2 in 0l2. We choose P, andP2 wo that they are both contained in some fundamental polygon E for J. LeaveP, fixed and define P2(t) = k,(P2), so that P2(t) is a fundamental polygon for F2(t)in the appropriate half-plane. By VII.C.2(vii), P, and P2(t) together bound afundamental domain D(t) for the action of G(t) on A(t) (see Fig. VIII.E.8).

There is also a homeomorphism tfi,: H2 - A(t) defined as follows. Choose asector S = {zI Iarg(z) - t/2l 5 S}, where 8 is chosen to be sufficiently small sothat S is precisely invariant under J in F. S divides D(to) into two regions: D,(to)in the right half-plane, and D2(to) in the left. Define 0, to be the identity in D,(t),to be a rotation by t in D2(t), and define it in S so that it preserves the modulus,and is a linear function of the argument. This defines a homeomorphism of D(to)

240 IX. B-Groups

onto D(t) that preserves the identifications of the sides. Hence we can extend 0,to a local homeomorphism of H2 into A(t). Since D, is a fundamental domain forthe action of G(t) on A(t), the image of H2 is all of A(t). The induced homomor-phism rp,: F -+ G(t), defined by (p,(f) = 0, ofo 0,7', acts as the identity on F1 andis an isomorphism of F2 onto F2(t) that preserves J; hence cp, is an isomorphism.It then follows that ti, is a homeomorphism of H2 onto A(t).

G.13. As t - t *, it is clear that cp, I F1 and 4p, I F2 both converge algebraically; thatis, on generators. Hence tp,: F - G(t) converges to a homomorphism cp: F - G.We chose po so that cp, is type-preserving for every t; hence by Chuckrow'stheorem (V.E.6), 4p is a type-preserving isomorphism.

It is also obvious from the way 4p, has been defined that as t -+ t*, gi,I D(to)converges to a homeomorphism of D(to) onto its image, D.

Lemma. G is Kleinian, and D is precisely invariant under the identity in G.

Proof. Let U be some nice open set in D(to)fl H2. Then U is also a nice open setin D(t) for every t e To. Then for every f # 1 in F, and for all t E T, (p,(f)(U) liesoutside D(t). Hence in the limit, (p(f)(U) lies outside D.

G.14. Let .4 be the component of G containing D. We can now define 0 mappingH2 onto A by using the fact that is defined on D(to) so as to preserve theidentifications of the boundaries. We then extend 0 to all of H2 by requiring thatiP -f(z) = (p(f) o bi(z). We know that Eli is a local homeomorphism; since 4, is anisomorphism, 4t maps ll2 homeomorphically onto some open invariant subsetof A. Let D* be the marked surface obtained by identifying the sides of D; thenD* is a toplogically finite Riemann surface, for it is homeomorphic to H2/F. Everynon-compact end of H2/F corresponds to a parabolic cusp on the boundary ofD(to). This cusp lies inside a cusped region C, where C is either contained inD1(to) or in D2(to). Then 0(C) is a cusped region for the corresponding parabolicsubgroup of G. Hence D* is a finite Riemann surface, and so its translates underG fill out the component A. We have shown the following.

Proposition. G is a B-group, with a simply connected invariant component, whereA/G is a finite Riemann surface homeomorphic to H2/F.

G.15. Lemma. G is not quasifuchsian.

Proof. Let A be the axis of J in A. Then A is a simple axis. Hence if G werequasifuchsian, the complete axis A would divide Q' into two closed discs, B1 andB2, where B,. is precisely invariant under J in This means that t* would liein To, which it does not.

G.16. In order to complete the proof of Theorem G.5, for the case that w isdividing, we need only show that G has no accidental parabolic transformations;


for we know that it is a non-elementary B-group, and not quasifuchsian (seeD.21). We constructed po so that if f c- F is not parabolic, then neither is cp(f ).Hence the only elements of G that are parabolic are the conjugates of theparabolic elements of F, and F2(t*). We saw above that these all representpunctures in A/G, so they are not accidental.

G.17. We turn now to the case that w is non-dividing. The proof is essentiallythe same as that given above, except that we use the second combination theorem,rather than the first. We give the proof in broad outline only.

Let A, be some axis lying over w, let J, = Stab(A 1), and let f, be the completeaxis of J,. Since w is non-dividing, there is a simple loop v that crosses w at exactlyone point z. Starting at some point lying over z on A, the lift of v determines anelement f e F. Let A2 = f(A,) and let J. = fJ, f -t.

Normalize F so that J2 has its fixed points at 0 and e, so that J, has its fixedpoints at I and x > 1, so that f(1) = 0, and so that f(x) = oo. Then, ignoring ourusual determinant condition, we can write f(z) = (bz - b)/(-z + x). Let B. bethe closed disc bounded by Am, where B, fl B2 = 0. Let I be the set of translatesof A, and A2 under F; let T be the region cut out by .E having A, and A2 on itsboundary, and let FO = Stab(T). Then B. is a strong (1m, Fo)-block, and Bt andB2 are jointly f-blocked. Also, there are many limit points lying between B, andB2 that are not translates of any point of either of these sets. Hence VII.E.5 isapplicable.

Let k,(z) = tz; write t = pei0. Let f, = k, of and let G(t) _ <Fo, f,>. For tsufficiently close to 1, it is easy to see that G(t) is quasifuchsian, and has thesame signature as does F. For every t, there is the obvious homomorphism:cp,: F - G(t).

Choose po so that for every 0, G(t) has no parabolic elements other than theimages under rp, of those of F, and their conjugates. Given po, let T be the set ofall t = poe1° so that the following hold. There are loops W, (t) and W2(t), whereWm(t) bounds a disc Bm(t) that is precisely invariant under tp,(Jr); also B, (t) andB2(t) are jointly f,-blocked.

Fix a fundamental polygon Do for To so that j,, a hyperbolic generator forJ,, identifies two of the sides of Do, and so that the axis A, of J, passes throughthese two sides; we can also require that a hyperbolic generator j2 for J2 identifiestwo of the sides of Do, and that the axis A2 of f2 passes through these two sides.Using VII.E.5, we obtain a fundamental domain D for F from Do and its reflectionin the real axis; note that D has two sides on A,, and two sides on A2.

For fixed t e T, the element f, maps the circle A, onto the line through theorigin: {arg(z) = n/2 + arg(t)}. This defines a fundamental domain D(t) for G(t),where the sides of D(t) are exactly the sides of Do, and its reflection, together withthe two arcs of A,, and the two line segments of {arg(z) = n/2 + arg(t)) thatthese are mapped onto by f,. This fundamental domain D(t) has two components,an upper component D, (t), and a lower component D2(t) contained in the lowerhalf-plane. As t moves along T (on the circle of radius po) in the positive direction,

242 IX. B-Groups

D,(t) is increasing, and D2(t) is decreasing. There is then a first point t* which isnot in T. Set G = G(t*). The proof that G is degenerate is very similar to theproof in the preceding case, and is left to the reader.

G.1S. We remark that this result can be significantly improved. Using standardmethods in quasiconformal mappings, one can find a degenerate group of anysignature as above, and any given conformal structure; that is, one can find adegenerate group representing any given marked finite hyperbolic Riemannsurface, other than a sphere with three special points.

G.19. Theorem. Let G be a degenerate group without accidental parabolic transfor-mations. Then G is not geometrically finite.

Proof. Assume that G is geometrically finite; we first take up the case that Gcontains a parabolic element j, with fixed point x. Let /3: 0 -+ H2 be a Riemannmap, and let F be the Fuchsian model of G. Since j is doubly cusped, it cannotbe accidental. Since every Euclidean subgroup of F is cyclic, Stab(x) is cyclic; wecan assume that Stab(x) = Q>. Since j is doubly cusped, there are two disjointcusped regions B and B' in Q(G); hence /3(B) and fJ(B') are two disjoint cuspedregions in H2 for Jt*(j)c- F; this is impossible. From here on we assume that Gcontains no parabolic elements.

We next assume that G contains an elliptic element j. Let A be the axis of jin H'. Since every elliptic element of F has exactly one fixed point in 0.82, thereis exactly one endpoint of A in 9, call it x. Let D be a finite sided fundamentalpolyhedron for G. Since G is geometrically finite and has no parabolic elements,the Euclidean closure of D is bounded away from A(G). Since only one endpointof A lies in Q, there must be infinitely many translates of D meeting A; i.e., thereis a sequence { g,,, } of elements of G, with all distinct, and g.(A) n b 0.Since G is discrete, any sequence of distinct translates of A is divergent in 1-8';that is, g.(A) converges to a point z on the Euclidean boundary of D. Since D isbounded away from A, z is necessarily in Q. We conclude that gm(x) - z, wherethe points g.(x) are all distinct, and z e 0. This cannot be.

Finally, we take up the case that G is purely loxodromic. Consider G as actingon S'. The limit set A(G) is still the same subset of C. Since its complement inC is connected and simply connected, its complement in §' is contractible. LetR = §' - A; i.e., R is the union of the upper half space, the lower half space, andQ(G). Then RIG is a closed orientable 3-manifold M, and we have S = Sl/Gembedded in M, and dividing M into two halves. Since R and 0 are bothcontractible, ttm(R) = nm(M) = 0, for m > 1. Hence R and M have the samehomology groups [34 pg. 201], and so they have the same Euler characteristic.Since the Euler characteristic of a closed orientable 3-manifold is zero [28 pg.283], S has genus 1. This contradicts the assumption that G is non-elementary.

0

IX.H. Groups with Accidental Parabolic Transformations 243

IX.H. Groups with Accidental Parabolic Transformations

H.I. Throughout this section, (G, A) is a non-elementary B-group with accidentalparabolic transformations. Denote the set of axes of accidental parabolic trans-formations by £; then £ is a set of simple disjoint loops, called structure loops.These loops divide A, and t, into regions. The regions in A are called structureregions; the stabilizers of the structure regions are called structure subgroups.

Let R be a structure region, and let H = Stab(R). Since £ is G-invariant, everyelement of G either stabilizes R or maps it to some other structure region R'.Hence R is an (H, G)-panel.

H.2. Let H be a structure subgroup of G. Since A(H) c A(G), Q(H) DO(G).Hence A is contained in a component, A(H) of H; A(H) is called the primarycomponent of H. Since H stabilizes A, H also stabilizes its primary component.

H.3. Proposition. Let W be a structure loop. Then there is at most one otherstructure loop W' with Stab(W) = Stab(W'). If W' 4 W and Stab(W') = Stab(W),then there is a structure region R where Stab(R) is infinite dihedral, Stab(W) is theparabolic subgroup of Stab(R), and W and W' are the only structure loops on OR.

Proof. Assume that there is a structure loop W' with Stab(W') = Stab(W). In theFuchsian model, if we have two true axes, A and A', with Stab(A) = Stab(A'),and Stab(A) is hyperbolic cyclic, then A = A'. Hence Wt and W2 are not the trueaxes, but the two axes on either side of the true axis. Since the stabilizer of thetrue axis is a Z2 extension of Stab(W), there cannot be a third axis between Wand W', so there is a single structure region R lying between them, and Stab(R)is equal to the stabilizer of the true axis, which, in this case is necessarily infinitedihedral.

H.4. Let R be a structure region, let H = Stab(R), and let A(H) be the primarycomponent of H. Each axis Won a(R) bounds a closed disc B, called the outsidedisc, that does not intersect R. Since W is precisely invariant under J = Stab(W)in G, it is also precisely invariant under J in H. Every element of H - J maps Wto another axis on the boundary of R; it follows that 9 is precisely invariantunder J in H. Since J is parabolic cyclic, A contains no limit points of H.

In general, the closed outside disc B is precisely invariant under J; there ishowever one case in which this is not so. Since there are no elliptic fixed pointson w fl 0, and p(W no) is a simple loop, w fl o is precisely invariant under J.Hence if W is not precisely invariant under J, then there is an element h e J, withW' = h(W) # W, but h(x) = x, where x is the fixed point of J. Since h(x) = x,J' = hJh-t commutes with J. Since every Abelian subgroup of the Fuchsianmodel F is cyclic, and J is a maximal cyclic subgroup of G, J' = J. By H.3, thisoccurs only when H is infinite dihedral.

244 IX. B-Groups

We have shown that if the closed outside disc is not precisely invariant underits stabilizer J, then the structure region R has exactly two structure loops on itsboundary, both are J-invariant, and H = Stab(R) is elementary.

H.5. Proposition. No structure subgroup is cyclic.

Proof. Let R be a structure region, let H = Stab(R); assume that H is cyclic. SinceG contains accidental parabolic transformations, there must be at least onestructure loop on OR. Let W be such a structure loop. Then J = Stab(W) isparabolic cyclic. A parabolic element that stabilizes a simple closed curve stabi-lizes both sides of the curve; hence J e H. In the Fuchsian model, the stabilizer ofan axis is a maximal cyclic group; hence J = H. By H.3, since Stab(R) is cyclic,there is no other structure loop W' with Stab(W') = J. Hence W is the uniquestructure loop on OR. Then both discs bounded by W are precisely invariantunder H in G, so G = H is elementary, contrary to our basic assumption that Gis non-elementary.

H.6. Proposition. Let R and R' be structure regions with Stab(R) = Stab(R'). ThenR = R'.

Proof. Suppose R 96 R'; let H = Stab(R) = Stab(R'), and let W be the structureloop on r3R between R and R'. Let 1) be the open disc bounded by W, and con-taining R'. Then A is precisely invariant under its stabilizer in H. Since R' c A,Stab(R') = H = Stab(W); this contradicts H.5.

H.7. Proposition. A structure subgroup H is elementary if and only if it is infinitedihedral; if H is elementary, then the structure region R stabilized by H has exactlytwo structure loops on its boundary, both stabilized by the parabolic subgroup ofH. Further, if W is a structure loop on OR, where Stab(R) is elementary, then thestructure region R' on the other side of W is stabilized by a non-elementary struc-ture subgroup.

Provj. A Kleinian group is elementary if and only if it is a finite extension ofan Abelian group. Hence if H is elementary, the Fuchsian model of H is alsoelementary. The only elementary groups that stabilize a disc are the cyclic groups,and 12 * 12. Hence the Fuchsian model of H is a 12 extension of a hyperboliccyclic group, so H is a 712 extension of a (n accidental) parabolic cyclic group.The structure region R, stabilized by H, has at least one structure loop W on itsboundary, and Stab(W) is the parabolic subgroup of H. For any half-turn h e H,h(W) = Wis also a structure loop on OR, with Stab(W') = Stab(W). By H.3, Wand W' are the only structure loops on OR.

If the stabilizer H' of an adjacent structure region were also elementary, thenit would also be a 12 extension of the same parabolic cyclic subgroup stabilizing

IX.H. Groups with Accidental Parabolic Transformations 245

the structure loops on its boundary. This would give three structure loops withthe same stabilizer, contradicting H.3.

H.8. Proposition. Let j be an elliptic element of G. Then there is a structure regionR, so that j has a fixed point in R; in particular, j lies in the structure subgroup,Stab(R).

Proof. Let fl: A - 6-02 be a Riemann map, and let F = f-'Gf be the Fuchsianmodel of G. Then j = f,(j) is an elliptic element of F, hence it has a fixed pointx in H2. Since there are no elliptic fixed points on axes of accidental parabolicelements of G, fl-'(x) lies in some structure region R. Since fl-(x) is a fixed pointof j, j(R) = R.

H.9. Proposition. Let j be a parabolic element of G. Then there is a structuresubgroup H c G, so that j e H.

Proof. Let fi be the Riemann map, and let F be the Fuchsian model of G, asabove. If j is accidental, then there is a j-invariant structure loop W, so j stabilizesthe structure regions on either side of W.

If j is not accidental, then there is a cusped region Ll c 0-02 for the maximalparabolic subgroup of G containing,] = f,(j). fl projects to a punctured disc onH2/F; since there are only finitely many equivalence classes of axes of accidentalparabolic transformations (D.19), we can choose R so small that it does notintersect any of the images of these axes. Then B = f-' (l) is a cusped region forj that is entirely contained in some structure region R. Then Stab(B) c Stab(R),so j e H = Stab(R).

The key fact in the proof above can also be proven directly. Normalize theFuchsian model F so that j(z) = z + 1. Let I be the true axis of f e F corre-sponding to an accidental parabolic transformation in G. Then A is preciselyinvariant under its stabilizer in F. Since no translate of A can intersect A, andj(z) = z + 1 lies in F, the (Euclidean) distance between the endpoints of T is notmore than 1. Hence I does not intersect { Im(z) > I).

H.1O. Lemma. Let H be a structure subgroup of G. Then (H, A(H)) is a B-groupwithout accidental parabolic transformations.

Proof. Let R be the structure region stabilized by H. Every element of G eitherkeeps R invariant, or maps it onto another structure region; i.e., R is preciselyinvariant under H in G. This is equivalent to saying that there is an opensubsurface Y c S = A/G, so that R is a connected component of p-'(Y). SinceR is defined by the axes of the accidental parabolic transformations, Y is boundedby the projections of these axes; this is a collection of simple disjoint loops. If Ghas signature (p, n; v1,..., v.), then there are at most 3p - 3 + n of these loops on

246 IX. B-Groups

S (see D.19), and any one of them can appear at most twice on the boundary ofR. We have shown that there are only finitely many H-equivalence classes of axeson OR.

Let W be an axis on the boundary of R, let J = Stab(W), and let B be theclosed outside disc bounded by W. Since J is parabolic cyclic, every point of Ais a point of d(H). Hence, if we take R and adjoin the regular points of B to it(that is, adjoin the points of B other than the fixed point of J), we obtain a setcontained in d(H). We do this for every axis on the boundary of R, and so obtaina set R; which is both H-invariant, and contained in d(H).

Combining the above statements, we obtain the following. R'/H is the sub-surface Y, with punctured discs (the images of the outside discs) sewn in at theboundary loops. Since there are only finitely many of these punctured discs, R+/His a finite Riemann surface. It follows from D.3 that R+ = d(H). Since R+ isH-invariant, (H, R+) = (H, d(H)) is a B-group; it remains to show that d(H) issimply connected, and that H contains no accidental parabolic transformations.

If V is any loop in d(H), then clearly, we can deform V in d(H) so that itdoes not pass through any of the axes on OR; in fact, we can deform it so that itlies entirely in R c A. Since d is simply connected, one side of V, call it the inside,contains no limit points of G. Then the inside of V surely contains no limit pointsof H. Hence V is contractible in A (H). We conclude that A (H) is simply connected;we have shown that H is a B-group.

Let j be some parabolic element of H. If j is accidental as an element of G,then the axis of j is a structure loop W on OR; W bounds a ball B whose interiorcontains no limit points of H. By D.15, j is not accidental in H.

If j is not an accidental element of G, then the image of j in the Fuchsianmodel of G represents a power of a small loop about a puncture. Hence there isa cusped region C c d, with vertex x, so that C is precisely invariant underStab(x). Then C is also contained in d(H), and is precisely invariant under Stab(x)in H. Here again, j is not accidental in H. D

IX.I. Exercises

U. Let G be the cyclic group generated by z -i kz, k > 1, and let S be the annulus1.12/G. Let won S be the projection of the positive imaginary axis. Express m(w, S)as a function of k.

1.2. Let a and b generate a triangle group G; i.e., G = <a, b: as = bo = (boa)' = 1 >.Call a homomorphism rp: G - PSL(2; 08) standard if sp(a) is elliptic of order a (orparabolic, if a = oc), Wp(b) is elliptic of order fi (or parabolic, if l3 = co), and qp(b o a)is elliptic of order y (or parabolic, if y = oo). For which or, /3, y is there a standardhomomorphism cp, where the image of qp is not discrete.

IX.I. Exercises 247

1.3. Construct a Kleinian group G, necessarily not analytically finite, and ahomeomorphism 0: Q(G) - Q(G), where 0 commutes with every element of G,but is not the restriction of a deformation. (Hint: Use VIII.A)

I.4. Let F, be a finitely generated Fuchsian group of the first kind, acting onH2, where j(z) = z + I is a primitive element of F,. Let h(z) = z + 4i, and letF2 = hF1 h-1. Then G, = <F F2> is a B-group, and j is accidental in G1. (Hint:use II.C.5, and VII.C.2.)

1.5. Let F be a Fuchsian group acting on H2, where F contains the primitiveparabolic elementj(z) = z + 1, and F is invariant under reflection in the imaginaryaxis (that is, conjugation by z - -z maps F onto itself). Let h(z) = - z + 4i.Then G2 = <F, h> is a B-group, j is accidental in G2, and the true axis of J con-tains elliptic fixed points. (Hint: Show that G2 is a 712 extension of the groupG, constructed in 1.4.)

I.6. For any Kleinian group G, call a parabolic element g weakly accidental ifthere is another Kleinian group C, and there is an isomorphism lp: G -+ t , sothat cp(g) is loxodromic.

(a) Which elementary groups contain weakly accidental parabolic elements?(b) Show that every parabolic element of a finitely generated Fuchsian group

is weakly accidental.

1.7. Let F be a Fuchsian group with two non-conjugate maximal subgroups ofparabolic elements, J1 and J2. Let C1 = C11 U C, 2 and C2 = C21 U C22 be doublycusped regions, for J, and J2, respectively, where (C1,C2) is precisely invariantunder (J1,J2). Let j;; be an element of M mapping the outside of C1, onto theinside of C2J, and let G,i = <F, f,;).

(a) Show that Gi; is Kleinian.(b) For which i and j is G0 a B-group?(c) Which pairs of these four groups {G;;}, i, j = 1, 2, are isomorphic?

I.8. Repeat 1.7 but starting with F a 712 extension of a Fuchsian group; that is, Fhas exactly two components, both circular discs, and neither invariant.

1.9. Let G be a Kleinian group with two invariant components, .4, and .42.Suppose G also has a third component d3. Then Stab(d,) is elementary.

1.10. Let S be a finite marked hyperbolic Riemann surface, and let { w1, ... , wk }be a system of loops on S. Construct a B-group (G,^ with a topologicalprojection p: A -' S, so that under the natural homomorphism, every wm cor-responds to an accidental parabolic transformation.

1.11. Construct a degenerate B-group that contains accidental parabolic trans-formations. (Hint: use a construction similar to that of 1.4, starting with adegenerate group.)

248 IX. B-Groups

1.12. Let G be a quasifuchsian group that is not Fuchsian. Use V.E.3 and V.G.18to show that the simple closed curve A(G) is not differentiable at a dense set ofpoints.

IX.J. Notes

A.I. A more extensive treatment of extremal length can be found in Ahlfors [5].A.3. Jenkins [35] showed that if rz, (S) is finitely generated, then m* (w, S) =m(w,S); see also Strebel [87]. A.4. [71]. A.5. Ahlfors [4]; a proof can also befound in Kra [43 pg. 324ff]. A.6. Yamamoto [101]; see also [70] and [71].C.I. There is a nice proof of this fact in the "folk literature", as follows. Sincethere is conformally only one sphere with three distinguished points, we canassume that the similarity f is conformal on 1-12. Then the Schwarzian derivativeof f is a quadratic differential on the sphere with three special points; hence it isregular everywhere except for at most simple poles at these three points. An easycomputation shows that zero is the only quadratic differential on the sphere withat most three poles. Since the Schwarzian derivative of f is zero, f e M. (Theinterested reader can find the necessary facts about quadratic differentials in Kra[43], and the necessary facts about Schwarzian derivatives in Duren [22].) CS.The fact that a hyperbolic triangle is determined by its angles can be found inBeardon [I I pg. 148]. D.21. [62], see also [ 15]. E.I. A proof of the Nielsenisomorphism theorem can be found in Marden [54], or Zieschang, Vogt andColdewey [104]; see also Tukia [92]. The original theorem is due to Nielsen[78]. F.I. See Thurston [90] for a study of quasifuchsian groups from the pointof view of hyperbolic 3-manifolds. The theorem of Bers is in [13]. F.6. Thistheorem first appeared in [62]; there is another proof by Kra and Maskit [45];see also Marden [52] and Thurston [90] for the torsion-free case. GA Degenerategroups were discovered on the boundary of Teichmiiller space by Bers [ 17], [15].G.B. This fact was observed by J.P. Matelski. G.19. This theorem is due toGreenberg [29].

Chapter X. Function Groups

This chapter has two main purposes; the first is to classify function groups up tosimilarity, and regular (i.e., geometrically finite) function groups up to deforma-tion, and the second is to show that every regular covering of a finite Riemannsurface, where the covering surface is planar, can be topologically realized by aregular function group. Using similar techniques with quasiconformal mappings,one can prove that every planar regular covering of a finite Riemann surface canbe conformally realized by a regular function group; this theorem however isbeyond the scope of this book.

One can define a function group either as a finitely generated Kleinian groupwith an invariant component, or, as we do here, one can define it as a Kleiniangroup G, with an invariant component d, where d/G is a finite Riemann surface.These two definitions are equivalent; it is almost immediate that the secondimplies the first, while one needs Ahlfors' finiteness theorem to go from the firstto the second. The finiteness theorem also has an extra dividend: one obtainsthat a function group, as defined here, is analytically finite. We prove this lastfact as a corollary of our decomposition of function groups.

X.A. The Planarity Theorem

A.I. In this section we give a description of all regular coverings of an orientablesurface, where the covering surface is planar.

Let X be a manifold, and let x0 be some point of X. One can describe anormal subgroup of rz, (X, xo) in terms of a set of loops (wm ) on X as follows.For each m, choose a spur v,,, from x0 to the base point of w,. Let a,,, bethe element of n, (X, x0) containing the loop vm w, Let N be the smallestnormal subgroup of n,(X,xo) containing all the elements {am}. We say thatN is the normal closure of the set { wm }, and we write N = <w1, w2,... >. Letp: (9, 20) - (X, x0) be the regular covering with defining subgroup N. It iseasy to check that N is independent of the spurs vm, and that this regularcovering is independent of all the base points; it depends only on the set ofloops {wm}. In this case, we say that p: 9 -+ X is the highest regular coveringof X for which the loops {wm} all lift to loops.

250 X. Function Groups

If the loop w on X lifts to a loop on A", then, with a slight abuse of language,we say that WEN.

A.2. A surface S is planar if it is topologically equivalent to an open connectedsubset of C. The Jordan curve theorem asserts that S is planar if and only if everysimple closed curve on S is dividing. An easy corollary of this is that a surface isplanar if and only if the intersection number of any two (homology) 1-cycles iszero.

Proposition. Let p: S - S be a regular covering, where S is a planar surface, andevery loop on S lifts to a loop on S. Then S is planar.

Proof. If S is not planar, then there are simple loops u and v, where u andv intersect at exactly one point so ES. Lift u and v starting at some point s"olying over so, to loops U and V, respectively. Since p is a local homeomorphism,U and V intersect, without multiplicity, at go. Since S is planar, there is asecond point of intersection s", # so. Since u and v intersect only at so, p(s,) = so.Hence the arc of U between so and s, projects to a loop that does not lift toa loop.

A.3. Proposition. Let S he an orientable surface, and let {wm} be a set of simpledisjoint loops on S. Let w,;, be the loop w,,, raised to some positive power. Let p: S' -+ Sbe the highest regular covering of S for which the loops {w., } all lift to loops. ThenS is planar.

Proof. Let W be a homotopically non-trivial loop on S. Choose base points goon S and so = p(go) on S; also choose spurs v, vm connecting so to the base pointof w = p(W), wm, respectively. Then v w v-' is homotopic to a product ofconjugates of the w,;,. Hence W is freely homotopic to a product of the form:r[ W, where for each k, p(Wk) is some vm wm v.'. Hence W is homologous tothe sum, Y W

For k # m, Wk n wm = 0, so any lifting of wk is disjoint from any lifting of w,,;hence the intersection in homology of any two such liftings is zero. If, for somem, V and V are both liftings of the same loop w' = w,;,, then since w = wm is simple,there is a simple loop w*, freely homotopic to w, where w and w* are disjoint(i.e., w* is "parallel" to w). Then the intersection number V x 17 is the same asthe intersection number V x V*, where V* is freely homotopic to V. and V*lies over w*. Since w and w* are disjoint, so are their liftings, V and V*. Itfollows that V x V 0. We conclude that for any two loops W and W' onWxW'=0.

AA. The converse of the above theorem is also true; we only prove it fortopologically finite surfaces. The proof given here is due to Gromov.

X.A. The Planarity Theorem 251

Theorem (the planarity theorem). Let p: S -+ S be a regular covering of thetopologically finite Riemann surface S, where ,9 is planar. Then there is a finite set{w,',,} of disjoint loops on S, where each w; is a power of a simple loop, so thatp: S - S is the highest regular covering of S for which the loops {w,, } all lift to loops.

AS. If S is simply connected, then we take the set of loops to be empty, and thereis nothing to prove. From here on, we assume that S is not simply connected.

The uniformization theorem tells us that the universal covering S of S, whichis also the universal covering of S, is either the sphere, the hyperbolic plane, orthe Euclidean plane. The sphere covers only itself, and is simply connected. If Sis the Euclidean plane, then S is either a torus, or the punctured plane (S is notthe plane, since S` is not simply connected).

If S is the punctured plane, then there is a simple loop w on S that generatesn i (S). Hence every regular covering of S is the highest regular covering for whichsome power of w lifts to a loop.

If S is a torus, then, since the universal covering of S is again the plane, theonly regular covering surfaces of S are the plane, the punctured plane, or anothertorus. If S is the punctured plane, which is the only case we need to consider,then G, the group of deck transformations on 9, is one of the elementary groupsconsidered in V.F. The only groups with two limit points, which act freely on allof Sl, are the loxodromic cyclic groups, and the groups 7L + 7.. Normalize anyone of these groups so that the limit points are at 0 and oo, and observe that theunit circle projects to a power of a simple loop.

From here on we assume, in addition to the assumption that S in not simplyconnected, that the universal covering of S is 0-U'. Our theorem is purely topologi-cal, so we can assume that S is the interior of a compact 2-manifold. This isequivalent to the assumption that the universal covering group is purely hyper-bolic. We will use the following two facts about hyperbolic surfaces. First, thereis a unique geodesic in each free homotopy class of loops on S (see V.G.8). Second,if two loops on S are disjoint then their corresponding geodesics are either equalor disjoint.

A.6. The main step in the proof of A.4 is an induction argument, which gives usa slightly stronger result. We state this as a separate theorem.

Theorem (Strong version of the planarity theorem). Let p: S -- S be a regularcovering of the hyperbolic Riemann surface S, where S is planar. Let N be thedefining subgroup for this covering. Suppose we are given simple disjoint loopsw1,. .. , wk on S, and we are given positive integers a 1, ... , ak, where a. is the smallestinteger for which w,- lifts to a loop, so that Nk = <w",..., w> e N, but Nk N.Then there is a simple loop w, disjoint from wt U U wk, and there is a positiveinteger a, so that w° e N - Nk.

A.7. Before going on to the proof of A.6, we observe that it implies A.4. For k = 0,A.6 gives us a first simple loop which, when raised to some power, lies in N (that


is, when raised to this power, it lifts to a loop). Then once we have found k disjointsuch loops, A.6 gives us a (k + 1)-st. Since we are on a hyperbolic surface, we canassume that each of these loops is a geodesic. We saw in V.G.15 that there areat most 3p - 3 + 2n simple disjoint geodesics on a hyperbolic Riemann surfaceof genus p with n boundary components. Hence, after a finite number of steps,this process terminates. At that point, we have found a finite set of disjointgeodesics {w;,..., wk}, where each w,;, is a power of a simple geodesic, andN = w,..... wk>.

A.8. Lemma. There is a homotopically non-trivial loop w' on S, where WEN,WO Nk, and w' is disjoint from w, U U wk.

Proof. Since N. # N, there is some loop v, where v c- N, and v 0 Nk. It is easy todeform v so that it intersects w, U . . U wk only at a finite number of simplecrossings. Let £k be the collection of liftings of the loops { w1, ... , wk}. Sincew,, ..., wk are simple and disjoint, so are their liftings. Since wk *k e N, c N, £k isa set of loops.

Let the loop V be some lifting of v. If V does not cross any of the loops in £k,we are finished; if it does, pick a loop A in £k that crosses V; A and V have onlyfinitely many points of intersection, and they are all simple crossings. Let s" beone of the points of intersection of V with A. Follow V pasts to the next pointof intersection s, with A. Pick one of the arcs U of A between s" and s,, and letU, and U2 be arcs that are close to U (i.e., they do not intersect any loop in £k),one on either side (see Fig. X.A. 1). V is freely homotopic to the product of twoloops; one follows V from nears to near s then follows Ul 1 back to the startingpoint near s'; the other follows U 1 in the opposite direction, and then follows V

Fig. X.A.1

X.A. The Planarity Theorem 253

Fig. X.A.2

from a point near s', to the other point nears (see Fig. X.A.2). Call these twoloops V, and V2, and let v1,, = p(V.).

Since V, and V2 are both loops, v, and v2 both lie in N. Their product is notin Nk, hence at least one of them is not in Nk. V, and V2 both have fewer pointsof intersection with 2k than does V (since the loops in 2 are simple and disjoint,we have introduced no new points of intersection, and we have eliminated thepoints of intersection at s and s`, ). Continuing inductively, after a finite numberof steps, we reach a loop W', which does not intersect any loop in 1k, where theprojection w' = p(W') lies in N - N. 0A.9. We now complete the proof of A.6. Since a set of loops on S is simpleand disjoint if and only if the corresponding geodesics are simple and disjoint,we can assume that the loops w,, ... , wk are all geodesics (in passing to geodesics,some of the loops might coincide, but that causes no difficulty). Let £ be theset of geodesics on S that lift to loops on 9, but do not cross any of the wm,and let w be the shortest element of E. There may be several shortest suchgeodesics, pick any one of them.

We first observe that if W is a loop in S lying over w, then W is simple. If not,then we could write W = V, V2, where v, = p(V,) and v2 = p(V2) are both in N(i.e., V, and V2 are both loops). Since we N - Nk, either v, or v2 also lies inN - N. They and their corresponding geodesics are both shorter than w.

Let W be the axis in H2 lying over w. w is a power of a simple loop on Sif and only if W is precisely invariant under its stabilizer in F, the Fuchsianuniversal covering group of S. This in turn occurs if and only if no translate oftf intersects If. Hence w is not a power of a simple loop if and only if w has adouble point, where the two tangents to the curve are not parallel.


Suppose w has a double point s. Let leg be some point of p-' (s), and letW, be the lift of w starting at s". Since s is a double point, there is also anotherlift W2 of w passing through s. Although W2 does not in fact start at s, repara-metrize it so that it does. Since the tangents to these two curves at s' are distinct,W2# W,.

Since S is planar, W, and W2 intersect at a second points",; since W, is simple,s, # s. Write W, = P - Q, W2 = R - S, where P and R go from s to s`, , and Q andS go from s`, back to s.

The arcs P, Q, R, and S all have well defined lengths, given by the Poincaremetric on H2, so that IPJ+IQI=IRI+ISI=IW11=IW21.

No one of the paths P, Q, R, or S can be shorter than the other three. Forexample, if P were shorter than all the others, then R - P-' and P - S would bothproject to loops that are shorter than w. Since W2 is homotopic to (R - P"') (P - S),at least one of these loops projects to a loop in N - Nk. Of course, the corre-sponding geodesic would be even shorter, contradicting the fact that w is theshortest geodesic in N - Nk.

The only other possibility is that IPI = IRI, or IPI = ISI. If necessary, replaceW, by Wi ', so that IPI 5 1 Wt 1/2. Also, if necessary, replace W2 by its inverse, sothat IPI = IRI. Now W, IPI + IRI =21PIsIW1I, and IR-QI=IRI+IQI=IPI+IQI=IW1I. Since s is an actualdouble point, the projections of these two loops have a corner at s (that is, thederivatives are discontinuous at s), so the corresponding geodesics must beshorter. Again, the projections of both these two loops have correspondinggeodesics that are shorter than w, and, since their product lies in N - Nk, at leastone of them lies in N - Nk.

A.10. It is important to note that the planarity theorem is a pure existencetheorem; in general, the set of loops {w,,..., wk} is not at all unique.

For example, let S be a closed Riemann surface of genus 2, and let(a,, b,, a2, b2 ) be a standard set of generators for n, (S) (see Fig. X.A.3 for oneexample of such a set of loops) Let c,, c2, and c3 be the geodesics on S freely

Fig. X.A.3

X.B. Panels Defined by Simple Loops 255

Fig. X.A.4

homotopic to b,, b2, and b, - b2, respectively (see Fig. X.A.4); these are simple anddisjoint. If we pick a base point so, then the subgroup of x,(S,so) generated byb, and b2 is the same as the subgroup generated by b, and b, b2, which is thesame as the subgroup generated by b2 and b, b2. The same statement is thentrue for the smallest normal subgroups; i.e., <b,, b2> = <b,, b, b2) = <b, b2, b2).An equivalent statement is the following. The highest regular covering of S forwhich any two of these three loops lift to loops is the same as the highest regularcovering for which all three lift to loops.

X.B. Panels Defined by Simple Loops

B.I. In this section, we consider the following situation. We are given a functiongroup (G, d); let S = d/G, and let °S = (d fl °Q)/G = °d/G. Note that °S is ananalytically finite Riemann surface. Let { w ... , wk} be a system of loops on °S(see V.G.15), and let Y be one of the building blocks cut out by the system. LetT be a panel lying over Y; that is, T is a connected component of p-1(Y). SetH = Stab(T), and note that T is precisely invariant under H.

Once we choose a starting point for the lift of w,, it determines an elementj. e G, where j,,, maps the starting point of this lift to the endpoint. While theelement jm is not uniquely determined by wm, its conjugacy class is. For a generalloop w, one expects the corresponding element j to be loxodromic; we specificallyrequire here that for each m, the element jm corresponding to a lifting of w beeither trivial, elliptic, or parabolic. Our final requirement is that if the elementj. is parabolic with fixed point xm, then StabG(x_) is required to have rank 1.

It follows from the planarity theorem, and the analysis of B-groups withaccidental parabolic transformations (IX.H), that this is a reasonable set ofrequirements.

B.2. Let w be one of the loops we let w be a lift of w, starting at some point onOT, and let j be the element of G determined by this lift. Set J = <j>.


If j is trivial, then w is already a loop, and w is precisely invariant under theidentity.

If j is elliptic, then j(iv) is a lift of w, starting at the endpoint of w; j2(w) is alift of w, starting at the endpoint of j(iv); etc. If a is the order of j, then w j(w) ...j°-`(w) is a loop; call it W. Since w is simple, and p is a local homeomorphism,W is a simple, J-invariant loop. We note incidentally that j is necessarily ageometric generator of J.

1ff is parabolic, look at the open arc: W° =It is clear that W° is precisely invariant under J. We can complete W° to a closedcurve W by adjoining the fixed point of J. Since p(W°) = w, W is simple.

In any case, we have produced a simple J-invariant loop W on aT. We callthis loop W a boundary loop of T; we say that W lies over w, even though theremight be a parabolic fixed point on it.

B3. Proposition. Let W be a boundary loop of T, let J = StabG(W), and letW° = W fl Q(G). Then J c H, and W° is precisely invariant under J in H.

Proof. The first statement is essentially immediate from the fact that S is orien-table. Every element of J preserves orientation, both as a transformation of C,and as a transformation of W. Hence every element of J preserves the two discsbounded by W; in particular, T is J-invariant.

Let he H, then h(W) is again a boundary loop of T, and h(W) and W lieover the same loop w. Since w is simple, either h(W°) = W°, or h(W°) fl W° = o.

Note that if J is parabolic, we might have that W and g(W) meet at the fixedpoint of J.

B.4. Proposition. Every boundary point of T either lies on a boundary loop, or is alimit point of H.

Proof. Let x be a boundary point of T, and let {xm} be a sequence of points ofT, with xm - x. Choose a subsequence so that p(xm) -' z. Since S is analyticallyfinite, z is either a puncture, a point on some w, or an interior point of Y.

If z is an interior point of Y, then there is a neighborhood U of p-` (z), andthere is a sequence of distinct elements {gm} of H with xmegm(U). Choose U sothat it is contained in T; since gm(U) fl T s 0, gm a H. Hence x = lim gm(p-`(z))is a limit point of H.

If z is a puncture, then there is a precisely invariant disc B c T, and there isa sequence of elements gm a G, with xm a gm(B), so that `(xm) converges to theparabolic fixed point on B. Since the sets gm(B) are all in T, the elements gm areall in H. If the sets gm(B) are all distinct, then gm(B) converges to some limit pointof H on 8T; then xm converges to the same point. If the sets gm(B) are all thesame, gm(B) = g(B), then the points xm converge to the g-image of the parabolicfixed point at the center of B.

X.B. Panels Defined by Simple Loops 257

If z is a point on some wk, then, as above, there is a neighborhood U of z,and there is a sequence of elements {gm} of H, so that xmegm(UflT). If theelements gm are all distinct, then gm(z) - x, so x is a limit point of H, while if thegm are all equal to some g, x lies on the boundary loop

8.5. Let W be a boundary loop of T. As a loop on t, W bounds two topologicaldiscs; the inside disc, which contains T, and the outside disc, which is disjointfrom T.

Proposition. Let B be the open outside disc of the boundary loop W. Then B isprecisely invariant under J = Stab(W) in H.

Proof. Every h c- H either stabilizes W, in which case h e J, or it maps W onto adisjoint (except perhaps for a parabolic fixed point) boundary loop of T Sinceh(W) cannot lie in B, h(B) fl B = o.

B.6. Proposition. Let W be a boundary loop, let J = Stab(W), and let B be theclosed outside disc bounded by W. Then B is a (J, H)-block. Further, B is strong ineither of the following circumstances: (i) J is not parabolic, or (ii) J is parabolic,and either G or H is geometrically finite.

Proof. We already know that 1 U W° = B fl S2(J) = B fl Q(H) is precisely in-variant under J in H. This shows that B is a block; it is of course strong if J isnot parabolic. Now assume that J is parabolic; let x be its fixed point. We knowthat Stabc,(x) has rank 1. If H is geometrically finite, then by VI.A.10, x is doublycusped in H. If G is geometrically finite, then x is doubly cusped in G, so of courseit is doubly cusped in H.

B.7. The subsurface Y is a connected component of the complement of S -{w, U U Looking at Y as a surface, each boundary component of Ycorresponds to one of the loops wm, and since locally, a loop has exactly twosides, each wm corresponds to at most two boundary components of Y. Hence Yhas at most 2k boundary components. We have shown that there are at mostfinitely many G-equivalence classes of loops on 3T.

Let W, and W. be G-equivalent loops on OT, that is, there is a g e G, withg(W,) = W2. Orient W, and W2 by choosing an orientation on w = p(W) =p(W2). Since W, and W2 both lie on the boundary of T, we can speak of them ashaving the same or opposite orientation as boundary loops of T. Since everyg e G preserves orientation, g e H if and only if W, and W2 both have the sameorientation as loops on 3T.

Pick a base point on each wm, and pick a base point on each boundary loop,so that the base points on the boundary loops lie over the base points on the wm.Then the loops W, and W2 on aT both project to the same boundary componentof Y (i.e., g e H) if and only if there is a path 6 in T, from the base point of W, to


that of W2, which projects to a loop v in Y. In any case v is a loop on S, so if v isnot a loop on Y (i.e., W, and W2 are not H-equivalent), then v crosses w on S atexactly one point. This can occur only is w is a non-dividing loop.

If w is non-dividing on S, and both sides of W are in Y, then there are loopsW, and W2, on the boundary of T, that project to different boundary componentsof Y. In this case, since there are only two choices for orientation, if W3 is a thirdloop on OT, W3 must be H-equivalent to either W, or W2.

We have shown the following.

Proposition. There are finitely many H-equivalence classes of boundary loops.

B.8. Let T' be the union of T, the boundary loops of T, excluding any parabolicfixed points that may lie on them, and the open outside discs bounded by theseboundary loops. It is immediate that T' is H-invariant.

Let Y' = T'/H. Since T is H-invariant, Y is a subsurface of Y'. There arefinitely many components of Y' - Y; one for each boundary component of Y(we saw above that these are in one-to-one correspondence with H-equivalenceclasses of boundary loops). Under the natural projection, each of these com-plementary components is the image of some closed outside disc B, together withits boundary, except perhaps for a parabolic fixed point. Since J = Stab(B) iseither trivial, elliptic cyclic, or parabolic cyclic, B/J is again a disc, perhaps withone special point in it. We conclude that T'/H is the surface Y to which we haveadjoined discs along the boundary components; in particular, T'/H is a finiteRiemann surface. Then II.F.8 yields the following.

Proposition. T' is a component of H, and (H, T') is a function group.

B.9. Proposition. Every loop in T' is freely homotopic in T' to a loop in T

Proof. Every component of T' - T is either a closed disc, or a closed disc withone point removed from the boundary. In either case, it is clear that any loopcan be deformed so as not to intersect such a set. 0

X.C. Structure Subgroups

C.I. Let (G, d) be a function group, and let S be the analytically finite markedRiemann surface d/G. Let °d = d fl °Q, and let °S c S be the image of °d underthe natural projection p. Suppose that d is not simply connected. By the planaritytheorem, there is a finite set {wm} of disjoint simple closed curves on °S, so thateach wm, when raised to some power, lifts to a loop, and p: °d - °S is the highestregular covering of °S for which these loops, when raised to these powers, lift toloops.

X.C. Structure Subgroups 259

If °S is the punctured plane, then we need only one loop in this set; call it wt.If °S is not the punctured plane, then we choose our set of loops {wm} in thefollowing fashion. First, there are q special points xt, ..., xq with finite markingon S; for each x, choose a small loop w,n about Note that each wm, whenraised to some power, lifts to a loop. Having found wt, ..., wq, we then find asystem of loops [w ...... w k } on °S, so that { lvt , ... , 1vq, wt , ... , wk } satisfies theconclusion of the planarity theorem; that is, p: 'd -'S is the highest regularcovering of °S for which these loops, when raised to certain powers, lift to loops.The loops w1,..., wk are called the preliminary dividers on S.

We also assume that the set { wt , ... , wk } is minimal. That is, if { v 1 , . . . , vk.) isa proper subset of {wt,..., wk}, then the highest regular covering of S, for whichthe loops wt, ..., wq, vt, ..., vk., each raised to the minimal power for which itlifts to a loop, is in fact higher than the covering p: °d - S.

We can also state this minimality condition in algebraic terms. If we let P.be the minimal positive power of wm that lifts to a loop, and let a, be the minimalpositive power of wm that lifts to a loop, and we look at the normal closure inrrt (°S) of some proper subset of {(wt) ...... (wk)°k }, where this proper subset con-tains all the then this normal subgroup is properly contained in N =

the defining (normal) subgroup for the covering p: °d °S.

C.2. Each connected component of the preimage of a preliminary divider is asimple loop, called a preliminary structure loop. Each preliminary structure loopprojects onto, or lies over one of the w, For each preliminary divider w there isa smallest positive integer a, so that w° lifts to a loop. Then, for every preliminarystructure loop W lying over w, there is a cyclic subgroup J of G, where J hasorder of, and W is J-invariant. In this case w is sometimes called an of-divider,and Wan a-structure loop. Note that J might be trivial, in which case x = 1.

Since each preliminary divider is simple, each preliminary structure loop Wis precisely invariant under its stabilizer J; hence each preliminary structure loopW is a (J, G)-block; since J is finite, it is strong.

C3. Proposition. Let be a sequence of distinct preliminary structure loops.Then dia(W,,,) -+ 0.

Proof. Since there are only a finite number of preliminary dividers, we can assumethat the W. all lie over one preliminary divider, w. That is, there is a sequence(g,,) of distinct elements of G so that W. = g,,,(W). The result now follows fromthe remark above and VII.B.14.

C.4. Since distinct preliminary dividers are disjoint, preliminary structure loopslying over different preliminary dividers are disjoint; hence the set of preliminarystructure loops is a set of simple disjoint loops. These divide d into regions,called preliminary structure regions. Each preliminary structure region lies overa component of S - {w...... wk}; hence the results of the preceding section areapplicable.


Proposition. Let x be a limit point of G. If there is a sequence {W.} of preliminarystructure loops which nests about x, then x is a point of approximation.

Proof. Choose a subsequence so that there is a single preliminary structure loopW, and there is a sequence { g,,,} of distinct elements of G, with W. = gm(W). Then,for all in, gm' (Wm) = W separates gm' (x) from gm' (W, ). Since W contains no limitpoints of G, and g,'(x) can accumulate only at limit points, gm'(x) is boundedaway from gm' (W, ). Then gm' (x) is bounded away from gm' (z), for any point zon a spanning disc for W, .

C.5. Let R be a preliminary structure region, and let H = Stab(R); H is called apreliminary structure subgroup. As in IX.H.2, we define the primary componentd(H) to be the component of H containing A. Since A is H-invariant, A (H) is alsoH-invariant.

The preliminary dividers divide S into building blocks. Each connectedcomponent of the inverse image of one of these building blocks is a subset of .4bounded by limit points of G and by preliminary structure loops (see B.4). Also,no preliminary structure loop intersects this set; hence these building blocks arethe projections of the preliminary structure regions.

C.6. Proposition. Let R be a preliminary structure region with stabilizer H. Then(H, A(H)) is a B-group.

Proof. We already know from B.8 that (H, A(H)) is a function group; we still haveto prove that d(H) is simply connected. Let V be a loop in d(H). By B.9, V isfreely homotopic to a loop in R; hence we can assume that V c R. Then p(V) = vis a loop on S. Choose a base point in S, and spurs to the preliminary dividers.Since v lifts to a loop, v is homotopic to a product of conjugates of preliminarydividers, and small loops about special points, all raised to various powers. Thelift of an appropriate power of a small loop about a special point with finiteramification number is a small loop about an elliptic fixed point in Q; hencecontractible in Q. Lifts of powers of preliminary dividers either lie in outsidediscs, or bound outside discs. In any case, since the closed outside discs areentirely contained in d(H), they are also contractible in d(H). Hence V ishomotopically trivial.

C.7. Proposition. Every elliptic element of G lies in a preliminary structuresubgroup.

Proof. Let j be an elliptic element of G. Suppose there is a fixed point x of j in A.Since the preliminary structure loops all lie in °d, x lies in some preliminarystructure region, and, necessarily, j keeps that preliminary structure regioninvariant.

If j has no fixed point in d, then define the j-number, # (j, R) of a preliminarystructure region R to be the minimal number of preliminary structure loops a


path between R and j(R) must cross. Let Ro be a preliminary structure regionwhich minimizes # (j, R); assume that # (j, Ro) > 0. Let V be a path from somepoint z e Ro toj(z), where V crosses the minimal number of preliminary structureloops. The paths j°(V), For a = 1, 2, ..., up to the order of j, fit together to forma closed curve; call it V'. Of necessity, there is a preliminary structure region R,so that V' enters and leaves R, by crossing the same preliminary structure loopW, and V' crosses no preliminary structure loop between these two crossingsof W.

Let R2 be the preliminary structure region V' leaves before it first enters R,;let x, be the first point of crossing of V' and W, and let x2 be the second, so thatthe arc A of V' between x, and x2 lies entirely inside R1. Sincej(R1) 0 R1, A liesentirely inside a fundamental set for the action of J on V'; that is, no two pointsof A are Q>-equivalent. Choose some arc A', lying entirely in R2 and connectingx1 to x2. Let V' be the loop V, with A, and its <j>-translates, replaced by A' andits <j>-translates. Since distinct preliminary structure loops are disjoint, thepoints x1 and x2 are not Q>-equivalent, so we can modify V' near x, and x2 sothat it does not even touch W at or near these points; call the new loop V *. Thenew loop V * is < j>-invariant, and has fewer crossings of preliminary structureloops than does V. Hence there is an Ro with # (j, R0) = 0.

C.S. Proposition. Every parabolic element of G lies in a preliminary structuresubgroup.

Proof. Let j be a primitive parabolic element of G. Choose some point x e A,where x does not lie on any preliminary structure loop, and draw a path A fromx to j(x). Extend A to a j-invariant arc V', and adjoin the fixed point x of j to V'to obtain a loop V. Assume that A passes through a preliminary structure loopW. Then V and W cross at a second point; let A' be the arc of V', lying betweenthe crossings with W. Since j(W) fl W = 0, j(A') fl A' = 0. Then, as in theargument above, we can change V' so that instead of of crossing W, it runs parallelto W. After this change, we have a new j-invariant loop, which we again call V,where this new loop has fewer crossings of structure loops, modulo J, than theoriginal loop did.

After a finite number of steps as above, we arrive at a j-invariant loop Vwhich crosses no preliminary structure loop. Then V lies in some preliminarystructure region, and Q> > = Stab(V) lies in the corresponding preliminary struc-ture subgroup.

C.9. Proposition. If j e G stabilizes two distinct preliminary structure regions, thenj is either elliptic or the identity.

Proof. If j stabilizes both R1 and R2, then j stabilizes the preliminary structureloop on OR, lying between them.


C.10. Let R be a preliminary structure region, and let H = Stab(R). Since(H, A(H)) is a B-group, H is either elementary, quasifuchsian (including Fuch-sian), or degenerate, or H contains accidental parabolic transformations(IX.D.2I ). We now take up the case that H has accidental parabolic transforma-tions; this cannot occur if H is elementary or quasifuchsian (see IX.D.9,17), butcan occur if H is degenerate. Let A c A(H) be the axis of an accidental parabolictransformation; as in IX.D, the axis is defined so that it projects to a simple closedcurve w on d(H)/H, and so that J = Staby(A) is cyclic.

It might be that w passes through some number of preliminary dividers.Recall that the preliminary structure loops on the boundary of R project to smallloops about ramification points in d(H)/H. Hence we can deform w on d(H)/Hso that it does not meet any preliminary divider. That is, we can deform A sothat, except for the parabolic fixed point, it lies entirely in R, and so that thedeformed curve is still precisely invariant under J. In fact, we can simultaneouslydeform all the axes of accidental parabolic transformations so that the deformedcurves are all still simple and disjoint, they all lie in R, and each deformed curveis precisely invariant under the parabolic cyclic group that stabilizes the originalaxis.

We adjoin the appropriate parabolic fixed points to these deformed curvesto get a set of simple disjoint loops which lie, except for the parabolic fixed points,entirely in R. These loops are called the oo-structure loops in R.

Since each preliminary structure region lies over one of the subsurfaces Y.,there are, up to G-equivalence, only finitely many preliminary structure regionsin A. Let R I, ..., R, be a complete list of non-equivalent preliminary structureregions. In each one of them we rind a finite set of non-equivalent oo-structureloops as above, and then we use the action of G to define the full set of oo-structureloops, so that this set of loops is G-invariant.

The full set of structure loops is the set of all a-structure loops, 1 5 a 5 co.Note that the set of structure loops is a set of simple disjoint loops which, exceptfor the parabolic fixed points on the oo-structure loops, lie in J.

The projection of the set of structure loops is then a system of loops {w...... wk}(different k) on S, called dividers. The dividers divide S into building blocksY1 , ... , YQ (different q).

Each connected component of the preimage of a building block is called astructure region; these are the regions cut out of A by the structure loops; as withthe preliminary structure loops, the set of structure loops is a set of simple disjointloops in C, which divide t, and A, into regions. The structure regions are preciselythe regions cut out of d by the structure loops.

C.11. Since the preliminary dividers are in general not unique (see A.10), neitherare the preliminary structure loops. For a B-group, the structure loops areuniquely determined, except that if there is a half-turn which stabilizes the trueaxis, then the structure loop is determined only up to free homotopy. For ageneral function group, in the presence of at least two preliminary dividers ond/G, the cc-structure loops are also not unique, even up to free homotopy.


The stabilizer of a structure region R is the corresponding structure subgroupH. Note that R is an (H, G)-panel.

We sometimes need to specify that a structure subgroup, or structure loop,or structure region is defined in terms of G, rather than in terms of some subgroupof G. We will sometimes refer to these as G-structure subgroups, or G-structureloops, or G-structure regions, to make clear the frame of reference.

Proposition. Let W be an oo-structure loop stabilized by the parabolic element j,and let x be the fixed point of j. Then StabG(x) has rank one, and every parabolicelement of StabG(x) lies in Stabo(W).

Proof. By C.8-9, there is a unique preliminary structure subgroup, H' = Stab(R')containing any parabolic element of G, in particular, j. Since j stabilizes astructure loop, it is accidental in H'. Since H' is a B-group containing anaccidental parabolic transformation, it is non-elementary. Since the Fuchsianmodel of H' contains no rank 2 Abelian subgroup, neither does H'. Of course,the fixed point x of j lies on 8R', and on the boundary of no other preliminarystructure region, so if j' is any element of Staba(x), then f e Stab(R') = H'. Weconclude that Stabo(x) = Stabn.(x) has rank 1.

One easily sees that if W is a simple loop that is precisely invariant under aparabolic cyclic subgroup J of some Kleinian group G, then J is a maximal cyclicsubgroup of G (see V.I.11).

We now know that the set of dividers on S satisfies the requirements of B.1;hence we can use the results of X.B.

C.12. Each structure region R is contained in a preliminary structure region R',and the corresponding structure subgroup H = Stab(R) c H' = Stab(R').

Proposition. Let H' be a preliminary structure subgroup of G, and let H be anH'-structure subgroup. Then H is a G-structure subgroup.

Proof. Once we have chosen the preliminary dividers, the B-group H' is welldefined. Once we have H', the accidental parabolic transformations are welldefined, but the corresponding oo-structure loops need not be. We need to showthat the structure subgroup H is well defined, independent of the choice ofoo-dividers. We know that there is a precisely invariant oo-structure loop foreach maximal cyclic subgroup of accidental parabolic transformations. For afixed such cyclic subgroup J, any two precisely invariant loops project to freelyhomotopic loops on d(H')/H', and so lift to homotopic loops based at the fixedpoint of J; in particular, there are no limit points of H' between them (but theremight be limit points of G between them). Then for any choice of co-structureloops in d(H'), the regions cut out by these loops have the same limit points ofH' on their boundary. It follows that the H'-stabilizers of these regions do notchange as we vary the ac-structure loops.


C.13. Proposition. Let H be a G-structure subgroup. Then H is a B-group withoutaccidental parabolic transformations.

Proof. We know from B.8 that H is a function group. Let R be a structure regionstabilized by H, and let R' be R with all outside discs adjoined, as in B.8. ThenR" = A (H), and every loop in A (H) can be deformed to a loop in R. For anyloop V in R, p(V) is a loop which lifts to a loop; hence p(V) is freely homotopicto a product of powers of small loops about special points, with finite ramificationnumber, and powers of preliminary dividers. The lift of an appropriate power ofa small loop about a special point with finite ramification number is a small loopabout a point of Q. Lifts of powers of preliminary dividers lie in outside discs,or bound outside discs lying entirely in d (H). Hence every loop in d is homotopicto a point; this shows that H is a B-group.

Suppose there is an accidental parabolic transformation j e H. Let H' be thepreliminary structure subgroup containing H, and let R' R be the preliminarystructure region so that Stab(R') = H'. Since j is accidental in H, both discsbounded by the axis of j contain limit points of H. These are also limit points ofH', so j is also accidental in H'. Hence there is an oo-structure loop W in R',stabilized by j. Since j stabilizes R. the fixed point x of j lies on OR. For the samereason, x e W. If W lies on OR, then by IX.D.15, j is not accidental in H. If Wdoes not lie on OR, then there is a structure region A between R and W. It isclear that every structure loop lying between R and W is also stabilized by j;in particular the structure loop on OR between R and W is stabilized by j. ByIX.D.15, j is not accidental in H.

C.14. A structure subgroup H is a B-group without accidental parabolic trans-formations, hence it is either elementary, quasifuchsian, or degenerate. If H isquasifuchsian, then it has two components; the primary component 4(H) -D A,and the secondary component 4'(H), disjoint from d.

Proposition. Let H be a quasifuchsian structure subgroup of the function group G.Then 4'(H) is a component of G.

Proof. Every point of 4'(H) lies in the exterior of d, and every limit pointof G lies on 84. Hence 4' c Q(G). Of course 4(H) c 4(G), so 4' is actually acomponent of G.

C.15. Proposition. Every non-loxodromic element of G is contained in somestructure subgroup.

Proof. Let g be a non-loxodromic element of G. By C.7-8, g lies in somepreliminary structure subgroup H'; then by IX.H.8-9, g lies in an H'-structuresubgroup H. By C.12, H is a G-structure subgroup.


C.16. Proposition. Let W be a structure loop, where J = Stab(W) is non-trivial.Then J is a maximal cyclic subgroup of G.

Proof. Suppose not. If J is elliptic, then there is a fixed point of J in each of thediscs bounded by W. Since J', the maximal cyclic subgroup of G containing J,has the same fixed points, for every j' E J' - J, j'(W) fl woo. This contradictsthe fact that W is precisely invariant under J.

Similar remarks hold if j is parabolic (see V.I.11).

C.17. It is immediate from the definition that every a-structure loop with stabilizerJ is a (J, G}block. The block is automatically strong if at < oc. If a = oc, the blockis strong if and only if J is doubly cusped in G.

Proposition. If {Wm} is a sequence of distinct structure loops of G, thendia(W,) - 0.

Proof. Since there are only finitely many G-equivalence classes of structure loops,the result follows at once from V1I.B.14.

C.IS. Proposition. Assume that every G-structure loop is a strong block, and let xbe a limit point of G. If there is a sequence { W.) of structure loops nesting aboutx, then x is a point of approximation.

Proof. Up to G-equivalence, there are only finitely many structure loops; hencewe can assume that there is a single structure loop W = W, , and there is asequence {gm} of elements of G, so that W. = If W is a preliminarystructure loop, see C.4. If W is an oo-structure loop, then observe that sinceeach W. separates x from W, W separates gm' (x) from gm'(W). Choose jm t= J =Stab(W) so that zm = hm(x) = jm o gm' (x) lies in a constrained fundamental set Dofor J. Since { Wm} nests about x, x is a limit point of G. Since W is strong, thepoints zm are bounded away from W. Of course, W separates hm(x) fromso the distance between hm(x) and hm(W) does not go to zero. Likewise, for anyz on a spanning disc for W, hm(x) and hm(z) are uniformly separated.

C.19. Proposition. Suppose R, and R2 are distinct G-structure regions. LetH. = Stab(Rn). Let W. be the structure loop on the boundary of R. separatingR1 from R2, and let J. = Stab(Wm). Then H, fl H2 is trivial unless J1 = J2. Also,ifJ,=J2,then H1f1H2=J,=J2.

Proof. Let It be some element of H1 - J, . Then, except perhaps for a parabolicfixed point on W, , h(W,) is disjoint from W, . Since W, separates R 1 from R2,h(W,) separates h(R1) = R, from h(R2); i.e., if h is not in J1, then R2 andh(R2) lie in distinct complementary components of R,. Hence H2 fl H, c J, =Stab(W1). Similarly, H1 fl H2 C J2. Since J1 and J2 are both maximal cyclic


subgroups of G, either J, fl J2 = (1), or J, fl J2 = J, = J2. In the former case,H, fl H. = {1 }; in the latter case, since J. c Hm, H, fl H2 = J, = J2. 0

C.20. Theorem. Suppose there is a structure region R where H = Stab(R) is ellipticcyclic of order v. Then the following statements are equivalent.

(1) There is a structure region R' # R, with Stab(R') = H.(ii) There is a structure loop W on OR, where Stab(W) = H.(iii) There are exactly two structure loops on OR stabilized by H.(iv) There is a primitive loxodromic element f E G that commutes with every

element of H.Also, if one, and hence all, of these conditions hold, then for every structure

region T, where H c Stab(le), there is an integer a so that k = f °(R).

Proof. Suppose first that (i) holds; that is, there is a structure region R' # R, withStab(R') = H. Let W be the structure loop on the boundary of R between R andR'. Then by C.19 H = Stab(W).

Now assume (ii). Since H is finite, A(H) = 0, and there are only finitely manystructure loops on OR; this means that R is bounded exactly by the finite numberof structure loops on its boundary. Also if W' # W is a structure loop on OR,then since Stab(W') is a maximal cyclic subgroup of G, either Stab(W') = {1},or Stab(W') = H.

Observe that every v-structure loop on OR separates the fixed points of H,while the i-structure loops do not. It follows that either there are exactly twov-structure loops on OR, which is what we want to prove, or there is only W, andH has exactly one fixed point z in R, which is what we now assume. SinceStab(R) = H, no other elliptic fixed point can lie in R. Let w' be a small loopabout p(z), and let W' be the connected component of p-`(w') near z; then W' isa small loop about z. Adjoin spurs to W, W', and the finite number of structureloops on OR. Let R' be R with the outside discs of the 1-structure loops adjoined.In R, W and W' are homotopic (or W and (W')-` are homotopic). We cannotexpect to project this homotopy to °S; but observe that the 1-dividers are trivialin it, (°S)/N, where N is the defining subgroup of the covering p: °d -* °S. Hencew = p(W) and w' are equivalent in it, (°S)/N. N is defined as the smallest normalsubgroup containing certain powers of certain simple disjoint loops; these loopswere chosen as first, small loops about ramification points, including w', andsecond, a minimal set of others needed to define N. We have shown that w is notneeded for this definition; i.e., we have contradicted the minimality of the numberof dividers.

We next assume (iii); let W and W' be the v-structure loops on OR. Theargument above shows equally well that p(W) and p(W') project to loops on °Sthat are equivalent in n, (°S)/N. Since there are a minimal number of preliminarydividers, this can occur only if p(W) = p(W'), up to orientation. Equivalently,there is an element f e G, with f(W) = W'. Since f(W) 96 W, f 0 H. Hence f(R) isthe structure region on the other side of W' from R. It is now immediate that f


is loxodromic. Also f conjugates H into itself; so either f commutes with everyelement of H, or f conjugates every h e H into its inverse. Since f is loxodromic,the latter case can not occur.

Continuing with this case, let F be the set of structure regions of the formf '(R), a e Z. Since f is loxodromic, lima +,,, dia(f'(W)) - 0. Since the structureloops between any two structure regions stabilized by H is H-invariant, and thereis no third H-invariant structure loop on the boundary of R, F is the full setof structure regions stabilized by H; that is, if k is a structure region withH c Stab(le), then k = f °(R) for some a.

Finally, we assume (iv) and prove (i). Then there is a primitive loxodromicelement f e G that commutes with every element of H. Since f # H, f(R) 0 R, andStab(f(R)) = fHf -' = H.

C.21. Proposition. Let H be a subgroup of the preliminary structure subgroup H'.Then H is a G-structure subgroup if and only if H is an H'-structure subgroup.

Proof. We know from C.12 that if H is an H'-structure subgroup, then it is aG-structure subgroup.

Assume that H is both a G-structure subgroup, and a subgroup of H'. LetR be the structure region stabilized by H, and let R' be the preliminary struc-ture region stabilized by H'. If R fl R' # 0, then R c R'; in this case H isan H'-structure subgroup. If R fl R' = 0, then by C.19, Stab(R) fl Stab(R') =H fl H' = H is contained in Stab(W), where W is the structure loop on OR'separating R' from R. We conclude from C.20 that H' = H. El

C.22. Proposition. Let R, 0 R2 be structure regions, where Stab(R,) = Stab(R2)-Then Stab(R I) is either trivial or elliptic cyclic.

Proof. It follows at once from C.19 that H = Stab(Rm) is either trivial, ellipticcyclic, or parabolic cyclic. If this last possibility were to occur, then there wouldbe a preliminary structure region R' containing both R, and R2. Then Stab(R')would be a non-elementary B-group, and Stab(Rm) would be a cyclic structuresubgroup of Stab(R'); this contradicts IX.H.5.

C.23. Proposition. Let R be a structure region, where Stab(R) = { 11. Then theidentity is the only structure subgroup, and all structure regions are G-equivalent.

Proof. Since Stab(R) is trivial, every structure loop on OR is a 1-structure loop,and R is bounded by these structure loops; that is, since R is precisely invariantunder the identity in G, R c °Q(G). Let W,_., W be the structure loops on R.Orient them so that they all have the same orientation from inside R. Choosespurs to a base point in R, then the product W, ... . W. is homotopically trivial.If say p(W,) were disjoint from the projections of W2,..., W,,, then p(W,) wouldbe a redundant divider. Hence for each W, on the boundary of R, there is a W,,


also on the boundary of R, and there is a g. E G, with gm(W.) = W.. Since theprojection of R, with these identifications of the boundary, is already closed(i.e., compact without boundary), it is all of S. Hence every structure region isG-equivalent to R, and the identity is the only structure subgroup (see X.H forfurther discussion of this case).

C.24. Proposition. Let H, 96 H2 be structure subgroups, and let R. be a structureregion stabilized by H,,,. If H1 fl H2 = J 0 111, then one of the following possibilitiesoccurs.

(i) There is a structure loop Won the common boundary of R, and R2; or(ii) J is parabolic cyclic; there is exactly one structure region R between R, and

R2; it is stabilized by an infinite dihedral group; the two J-invariant structure loopson DR are the only structure loops stabilized by J; and Stab(R and Stab(R2) areboth non-elementary: or

(iii) J is elliptic, and every structure region lying between R, and R2 has finitestabilizer.

Proof. We assume that the first case does not occur so that there is at least onestructure region R between R1 and R2. Let W,,, be the structure loop on theboundary of R. separating R, from R2, and let J. = Stab(Wm). By C.19, J, =J2 = J, and J is either elliptic or parabolic cyclic. R has two boundary structureloops separating R1 from R2 (these may or may not be different from W1 andW2). Since J stabilizes both W1 and W2, it also stabilizes every structure loopthat separates them; in particular, the two structure loops on the boundary of Rare J-invariant. Set H = Stab(R).

If J is parabolic cyclic, then by C.9, R1, R2, and R all lie in some preliminarystructure region R', with stabilizer H'. Then by IX.H.3, there are at most twostructure loops stabilized by J, there is a unique region R lying between W1 andW2, and R is stabilized by an infinite dihedral group. By IX.H.7, Stab(R 1) andStab(R2) are both non-elementary.

Assume next that J is elliptic cyclic. Since there are two J-invariant structureloops on OR, the primary component of H contains both fixed points of J. ByC. 13, H is a B-group without accidental parabolic transformations; hence it iseither quasifuchsian (including Fuchsian), degenerate, or elementary. The limitset of a quasifuchsian group separates the fixed points of every elliptic element;hence H is not quasifuchsian. Every elliptic element of a degenerate or Euclideangroup has exactly one fixed point lying in the limit set. The only possibility leftis that H is finite.

C.25. Theorem. Let (G, A) be a function group. A subgroup H of G is a structuresubgroup if and only if H is a maximal subgroup of G satisfying the followingconditions:

(i) (H, A(H)) is a B-group without accidental parabolic transformations,(ii) if x e A (H) is the fixed point of the parabolic element j e G, then j e H.


Proof. We first prove necessity. Condition (i) is just C. 13.Let j be a parabolic element of G, with fixed point x e A(H). By C.15, there is

a structure region R0, stabilized by the structure subgroup H0, where j e Ho. IfRo = R, then j e H; assume that Ro 0 R. Let Wo be the structure loop on aRoseparating it from R. Since the fixed point x of j lies in both OR and aRo, Wo mustbe an oo-structure loop, stabilized by j, and containing x. Similarly, the structureloop W on OR that separates R from Rp is an oo-structure loop containing x.Since x e W, j e Stab(W) c Stab(R) = H.

We now know that every structure subgroup satisfies these two properties;we still have to show that it is maximal.

Suppose there is a subgroup K c G, where K has the above two properties,and K properly contains H. Let k be some element of K - H. Since H =StabG(R), k(R) R. Let W be the structure loop on the boundary of R thatseparates R from k(R). There are now several cases to consider.

We first take up the case that H is non-elementary. Since A(H) # 0, andA(K) is connected, W can only be an oo-structure loop; let J = Stab(W). Sinceboth discs bounded by W contain limit points of K, by IX.D.12, J is acci-dental in K. This contradicts the fact that K contains no accidental parabolictransformations.

Next assume that H is rank 2 Euclidean. There is no isomorphism betweena rank 2 Euclidean group and any Fuchsian group, so K must also be a rank 2Euclidean group, with the same fixed point. Since the stabilizer of a parabolicfixed point on a structure loop has rank 1 in G (C.11), every structure loop onOR is a v-structure loop, v < oc. In particular, W c Q(K), and W separates thelimit point of H and from that of kHk-t.

Next assume that H is a rank I Euclidean group with fixed point x. Since Wdoes not separate limit points of K, W is an oo-structure loop; hence H is asubgroup of a non-elementary preliminary structure subgroup. Since StabG(x)has rank 1, and K properly contains H, K is non-elementary. By IX.H.5, H isnot cyclic. Since H is elementary, it is infinite dihedral; such a group cannot bea subgroup of any Fuchsian group, so the Riemann map from A(K) conjugatesH into Z2 * 7L2 (this is the group generated by two half-turns, where the productis hyperbolic). It follows that the parabolic subgroup of H is accidental in K.

If H is finite and non-cyclic, then H has a fixed point in I13. Every structureloop on OR has finite stabilizer, and so is a strong block. For each structure loopW' on OR, let C(W') be a spanning disc for W' (see VII.B.16). The set of all thesespanning discs, and their G-translates divides H3 into regions; it is easy to seethat the region 1(H) spanning R is precisely invariant under H. Hence the fixedpoint of H lies in I(R). It follows that C(W) separates the fixed point of H fromthat of k(H). Since K does not have a single fixed point in 0-03, it is not finite.Since neither Euclidean nor Fuchsian groups contain non-cyclic finite groups,we have reached a contradiction.

If H is elliptic cyclic, and there is no H-invariant structure loop on OR, thenboth fixed points of H lie in R, and the axis of H lies in I(R). Hence C(W) separates


the axis of H from that of k(H); as above, this implies that K is not finite. Bothfixed points of H lie in A(K), and for either a Fuchsian or Euclidean group, onlyone fixed point of each elliptic element lies in the invariant component; henceA(K) is neither hyperbolic nor parabolic; it follows that K is finite. We havereached a contradiction in this case.

Now consider the case that H is elliptic cyclic, and there are H-invariantstructure loops on OR. By C.20, there are exactly two such structure loops W,and W2 on 8R; also, there is an element f e G, where f has its fixed points at thefixed points of H, and f(W,) = W2. Since W, separates the fixed points of f, nopower off is in K. Then k(R) is not one of the structure regions of the form f '(R);in particular, k(R) is not stabilized by H. Note that if R' is a structure region, andthe axis of H passes through 1(R'), then R' is a structure region of the form f '(R).Hence, as in the preceding case, the axes of H and k(H) do not intersect, so K isnot finite. If K is Euclidean, then K has a fixed point at one of the fixed pointsof H. This cannot be, for f is loxodromic, and has fixed points at the fixed pointsof H (see II.C.6). If K is non-elementary, then one of the 1-structure loops on ORseparates the axis of H from the limit set of K; i.e., H has two fixed points inA(K); we saw above that this is impossible.

Finally we take up the case that H is trivial. Then every structure subgroupis trivial, and every structure region is G-equivalent to R. Hence every non-trivialelement of K is loxodromic (in fact, every non-trivial element of G is loxodromic),so K is non-elementary. There is a preliminary structure loop W, c 8k(R) whichseparates k(R) from k2(R), then there is a preliminary structure loop on theboundary of k2(R) which separates k2(R) from k3(R), etc. Hence there is a nestedsequence of preliminary structure loops which nest about the attracting fixedpoint of k. Hence the limit set of K is not connected.

C.26. We now prove the second half of C.25. Assume that K is a maximalsubgroup of G satisfying the two properties. There are again several cases toconsider.

We first take up the case that K is non-elementary. We noticed above thatfor every limit point x of G, either x lies on the boundary of some preliminarystructure region R, or there is a sequence of preliminary structure loops nestingabout x. Since A(K) is connected, there is some preliminary structure region R',so that A(K) c OR'. Since K stabilizes A(Stab(R')), K c Stab(R'). There cannotbe an cc-structure loop in R' which separates A(K), for then its stabilizer wouldbe accidental in K. Hence A(K) lies on the boundary of a single structure regionR; it follows that K c Stab(R). Since K is maximal, K = Stab(R).

If K is a Euclidean subgroup of G with fixed point x, then every parabolicelement k e K lies in some preliminary structure subgroup H(k), where H(k)stabilizes the preliminary structure region R(k). For every k e K, x e 8R(k), sothere is a single preliminary structure region R' with x e DR'. Then K C Stab(R').If Stab(R') is also Euclidean, then K and Stab(R') are both maximal Euclidean

X.D. Signatures 271

subgroups of G with the same fixed point, so they are equal. If H' = Stab(R') isnot Euclidean, then it is a B-group, and it contains some number of structuresubgroups. By IX.H.9, every parabolic element of K is contained in some struc-ture subgroup of H'; hence there is at least one structure region whose boundarycontains the fixed point of K, and thus whose stabilizer H contains the parabolicsubgroup of K. It follows that K has rank 1. Since K is maximal, K H; byIX.H.5, H is not cyclic, so K must be infinite dihedral. By IX.H.7, there are exactlythree structure regions stabilized by the parabolic subgroup of K, and there isexactly one structure region stabilized by K.

Since our conditions are necessary, if K is trivial, then every structure sub-group is trivial. If K is finite and cyclic, then our result follows from C.15. Finallywe take up the case that K is finite and not cyclic. Then every element k E K iscontained in some structure subgroup H(k). Since H(k) is maximal, it containsK, so K and H(k) are maximal finite subgroups of G with the same fixed pointin 0-13; i.e., K = H(k) for every k.

C.27. It is immediate that if R, and R2 are G-equivalent structure regions, thenStab(R,) and Stab(R2) are conjugate structure subgroups of G. The converse isalso true.

Proposition. Let R1 and R2 be G-structure regions, and let Hm = Stab(Rm). H1 andH2 are G-conjugate if and only if R1 and R2 are G-equivalent.

Proof. Assume there is an element g e G with H2 = gH1 g-1. Let R = g(R 1);observe that Stab(R) = H2. If R R2, then since H = Stab(R) = Stab(R2), H iseither elliptic cyclic, or trivial. The former case occurs only if there is a loxodromicelement f c- G, so that R2 = f '(R); in this case R 1 and R. are G-equivalent. Thelatter case occurs only if every structure region is G-equivalent to every otherstructure region.

X.D. Signatures

D.I. In this section we define the signature of a function group. The definitiondepends on some choices; we show that the signature is well defined; it dependsonly on the function group. We also write down some necessary conditions forthe signature of a function group; we show in the next section that these condi-tions are sufficient as well as necessary.

The signature of a function group consists of both a geometric object K, calledthe marked 2-complex, and a non-negative integer t, called the Schottky number.

The marked 2-complex contains two kinds of objects. First, there is a markedfinite (disconnected) Riemann surface X; the connected components of X are the


parts of K. There are also some 1-cells, called connectors. The end points of theconnectors are special points of X; the two endpoints of each connector havethe same ramification number. A connector connecting two points of order v iscalled a v-connector.

A theoretical signature is a marked finite (disconnected) Riemann surfaceX, a finite set of disjoint connectors {c1,...,ck}, and the Schottky number t,satisfying the following. Except for its endpoints, each c. is disjoint from X; thetwo endpoints of cm are distinct special points with the same ramification numberv, 2 5 v 5 co; and each special point of X is an endpoint of at most one connector.

We consider two signatures a = (K, t) and Q = (R, i) to be the same if thereis a homeomorphism from K onto R, where the homeomorphism preservesdistinguished points with their orders, and t = i.

D.2. We now start with a function group (G, d), and construct its signature.If (G,.4) is a B-group without accidental parabolic transformations, then K

is the marked surfaced/G, and t = 0.For any other function group, there is a finite set {w1,...,wk} of dividers on

S = d/G. If wm is a 1-divider, cut S along w., so as to produce two boundarycurves on this new surface, or surfaces, and fill in these two boundary curves withdiscs. Having done this, we obtain either a new connected surface, whose genusis one less than that of S, or a new surface with two connected components.

If w,a is a v-divider, v > 1, then we cut S along wm to again get a new surface,or surfaces, with two boundary components. Fill in each of these two boundarycomponents with discs, choose one point in each disc, and make it a special pointof order v. Join these two new special points of order v with a v-connector c.

After performing the above operations with each of the dividers w1,. .. , wk, wearrive at a possibly disconnected marked finite surface X, and a collection of con-nectors. This is the marked 2-complex K in the signature of (G, d). In Figures X.D.1through X.D.5, these operations are demonstrated on a closed surface of genus 5,where the dividers w,_., w4 are marked respectively with the numbers 1, 2, 1, 3.

Fig. X.D.1

Fig. X.D.2

Fig. X.D.3

Fig. X.D.4

X.D. signatures 273


D 3

Fig. X.D.5

Let r1 be the number of 1-dividers, and let ro be the number of connectedcomponents of K (two parts which are connected by a connector lie in the sameconnected component). Then the Schottky number t = T1 - ro + 1.

The operation defined by a v-divider, v > 1, in going from S to K, keeps Toinvariant; hence t >- 0. It is easy to see that t can also be defined as the maximumnumber of 1-dividers on S that can be cut open without dividing S; that is, t isthe maximal number of homologically independent 1-dividers.

D3. The 2-complex K has a well defined genus, obtained by "fattening" eachconnector into a handle, so that K becomes a marked surface. The genus of Kis the sum of the genera of the fattened connected components of K.

Proposition. The difference between the genus of S and the genus of K is the

Schottky number t.

Proof. Let w be a v-divider on S, v > 1. Cut S along w, sew in discs, pick a pointin each disc, and join these two points by a connector c. Now fatten c to obtaina surface, and observe that this surface is again S. Hence the difference betweenthe genus of S and the genus of K, if there is any, is entirely accounted for by the1-dividers.

Let w1 be a 1-divider on S. Cut S along w1, and sew in two discs to obtainK1. Either w1 divides S into two surfaces, in which case S and K1 have the samegenus, or w1 does not divide S, in which case there is a difference of I in the generaof S and K1. Hence t = rl - (To - 1) = genus(S) - genus(K). 0

D.4. The above operations can be performed all at once, instead of sequentially.That is, cut S along all the dividers, fill in discs, two for each divider, and if thedivider is a v-divider, v > 1, adjoin a v-connector from a point in one of thesediscs to a point in the other.

X.D. Signatures 275

The dividers divide S into building blocks; there is a one-to-one corre-spondence between the building blocks and the parts. Each part P is obtainedfrom the corresponding building block by filling in discs along the boundarycurves, and then marking at most one of the points in the disc. Hence, there is aninjection from the building block into the corresponding part P; we call the imageof this injection the real part of P, and the complement, consisting of the disjointclosed discs, is the imaginary part of P. We write these as Re(P) and Im(P),respectively. The discs in lm(P) are sometimes referred to as imaginary discs.

D.5. We next show that the signature of a function group is well defined inde-pendent of the choice of dividers.

Theorem. Let (G, A) and (ti, A') be function groups, and let (p: A be a type-preserving similarity with induced isomorphism cp,: G -+ C. Then

(i) cp, maps structure subgroups onto structure subgroups, and(ii) G and d have the same signature.

D.6. We start the proof of this theorem with the following.

Lemma. Let H be a structure subgroup of G, and let 13 = (p, (H). Then (17, d(g))is a B-group without accidental parabolic transformations, and 17 contains everyparabolic element of 0 whose fixed point lies in A(17).

Proof. Let R be a structure region stabilized by H. Set A = cp(R); since qp is asimilarity, A is precisely invariant under A? in C. Let W be a structure loop onOR, and let J = Stab(W). Then J= cp,(J) = Stab(W), where W = rp(W). Let B,R be the closed outside disc bounded by W, W, respectively. Since w fl A isprecisely invariant under J in H, W fl d is precisely invariant under 7 in R; henceA is precisely invariant under 7 in 13.

Form the set A+, as in B.8. Note that q establishes a homeomorphismbetween R/H and /7/R. Up to H-equivalence, there are only finitely many of theWon OR; hence we can obtain 1'/13 by adjoining a finite number of sets of theform (. fl e(R))/7 to RI!?, with the obvious identification of (W fl d)/7 as theboundary curve of RI!?. Since rp, is type-preserving, each 7is either trivial, ellipticcyclic, or parabolic cyclic; hence, as in B.8,R+/1l is a finite surface. It follows thatA+ = A(17'), and that (H,A(13)) is a function group.

Let V be a loop in d(1?). As in B.9, we can deform P so that it lies entirelyinside A, which is simply connected. Hence A' is a B-group.

Now suppose that j is a parabolic element of R. Then j = q -'(J7) is parabolic,and since H is a structure subgroup of G, j is not accidental in H. If H iselementary, then so is 1?, and no parabolic element of an elementary group isaccidental. If H is non-elementary, then there is a precisely invariant disc C c R.Then cp(C) is precisely invariant under j; hence j is not accidental in g. We haveshown that 1 is a B-group without accidental parabolic transformations.


Supposef is a parabolic element of C, and the fixed d-point x of f lies in A(A).Let j = cps' (f ). By C.15, j lies in some structure subgroup H'. Let R' be a struc-ture region stabilized by H'. If R' = R, then H' = H, which implies that je13.Assume that R' -A R, and that H' : H.

Let W be the structure loop on OR separating R from R'; let W = p(W), andlet A' = cp(R'). We know that f stabilizes IT' and that the fixed point x off lies onA(I?). Hence z lies in the intersection of the boundaries of . and k; i.e., W a Oft,lT' separates 1 from k, and x lies on 11 Since 1$' is not entirely contained in d,W = cp-' (l$') is an oo-structure loop; let x the the parabolic fixed point on W. ByC. 11, Stabo(x) = Stabu(W). Hence the maximal Abelian subgroup of C containingStab(W) is Stab(t'); it follows from this that f eStab(1P) c Stab(I) = 13. C3

D.7. Lemma. If H is a structure subgroup of G, then 17 = tp,(H) is a structuresubgroup of 0.

Proof. By D.6, ,R satisfies conditions (i) and (ii) of C.25. If 17 were not maximal,then it would be properly contained in a maximal such group R'; by C.25, 13' isa structure subgroup of C. Apply D.6 in the reverse direction to conclude thatH is not maximal. Since H is a structure subgroup, it is maximal.

D.8. Note that D.7 is conclusion (i) of Theorem D.5. To prove the second part,we first prove that there is a homeomorphism between the parts of K and R.

By D.7, there is a one-to-one correspondence between the structure sub-groups of G and those of 6. Then by C.27, there is a one-to-one correspondencebetween the parts of K and those of R. Let P be a part of K, and let P be thecorresponding part of R; that is, there is a structure region R, ff, lying over thereal part of P, P, respectively, and the isomorphism p* maps the structuresubgroup H = Stab(R) onto the structure subgroup R = Stab(I ). Since H and13 are B-groups without accidental parabolic transformations, there is a type-preserving isomorphism from each of them onto a Fuchsian or elementary group.Then by V.G.6-7, the corresponding Fuchsian or elementary groups have thesame basic signature; that is, there is a homeomorphism from P onto P, whichpreserves special points with their orders. In fact, we could choose this homeo-morphism so that its induced map on homotopy is that given by the projectionof cp. I H, but we do not need such a strong result. For our purposes, it sufficesto observe that there is a homeomorphism of any connected surface to itself,taking any set of n distinct points to any other set of n distinct points. Inparticular, we can assume that if the elliptic or parabolic subgroup J c H liesover a small loop about the special point x on P, then (p*(J) lies over a smallloop about the image of x in 13.

Unfortunately, the images of the special points on the parts of K need not beuniquely determined. It is clear that if H is Fuchsian, quasifuchsian, or degener-ate, then every elliptic element of H has exactly one fixed point in A(H), and everyparabolic element has exactly one equivalence class of precisely invariant circular

X.D. Signatures 277

discs in d (H), where two of these discs are in the same class if they have non-trivialintersection. The same statement is true for rank 2 Euclidean groups, but falsefor rank 1 groups; a parabolic cyclic group has two classes of precisely invariantdiscs. For a finite group H, there is a homeomorphism from d(H)/H onto itself,which lifts to a homeomorphism of d(H) that commutes with every elementof H, and which interchanges two special points on d(H)/H, if and only if Hhas signature (0, 3; 2, 2, v), v odd, or (O,3;2,3,3) (see V.C.7-8). This means thatif P has one of these signatures, we may not have chosen our homeomorphismcorrectly.

D.9. Our next goal is to show that the homeomorphism,f, from the parts of Kto those of R, can be modified so as to preseve endpoints of connectors. We startwith the observation that for each special point x on K, where x is the endpointof the connector c, there is a corresponding divider on S, or equivalently, thereis a corresponding boundary loop on the corresponding building block. Also,two special points x, and x2 on K have a connector between them if and onlyif the corresponding boundary curves on the boundaries of the correspondingbuilding blocks are in fact the same. This means that the corresponding buildingblocks have a v-divider, v > 1, as a common boundary loop. This in turn occursif and only if there are structure regions R, and R2 lying over the correspondingbuilding blocks, or parts, and there is a v-structure loop W on the commonboundary of R, and R2, v > 1, where the outside of W, from the point of view ofRm, projects to the imaginary disc containing xm.

Let c be a v-connector in K connecting the points x, and x2, where xm lieson the part P. (it is not excluded that P, = P2). Then there are structure regionsR, and R2, with stabilizers H, and H2, respectively, so that R, and R2 have astructure loop W as their common boundary, and the outside disc of W, viewedfrom the point of view of Rm, projects to the imaginary disc about xm. LetJ = Stab(W) = Ht fl H2; by assumption, J is a maximal elliptic or paraboliccyclic subgroup of G.

Let 17. = cp,*(Hm). These are structure subgroups of C, so there are struc-ture regions Rm, where Stab(Am) = 17m. Let 7 = q (J), and let rvm be thestructure loop on the boundary of Rm that separates k, from f2. Observe that.7 c R, fl l2; so 7 stabilizes both l$', and 1'2. This is equivalent to sayingthat the corresponding part m has a special point Xm of order v on it. It is notnecessarily true that f maps xm to zm, but, as we observed above, we can modifyf so that it does. If W, = a'2, then there is a connector between z, and i=,which is what we wish to prove. If not, then there are some number of structureregions between R, and K2, all stabilized by T. Let ft be one such structureregion; by C.24, 17 = Stab(f) is either finite or infinite dihedral.

D.10. We continue with our assumption that PV, # W2, and that there is a regionft between them. We first take up the case that Stab(f) is infinite dihedral, thenit is the only structure region between A, and f2, and both these regions have


non-elementary stabilizers. Then H = qp#-' (Stab(R)) is an infinite dihedral sub-group of G, and H is a structure subgroup different from both H, and H2. LetR be the (unique) structure region stabilized by H. We now have three distinctstructure subgroups, H, H,, and H2, all containing J. This means that there areat least three distinct structure regions stabilized by J, which in turn means thatthere must be at least two structure loops stabilized by J. Then by C.24, thereare exactly two structure loops and three structure regions stabilized by J, andthe (unique) such structure region stabilized by the elementary structure sub-group H lies between the other two. This contradicts our assumption that R,and R2 are adjacent. We have eliminated the possibility that Stab(A) is Euclidean.

D.11. We now take up the case that 7 is elliptic and R is elliptic cyclic. Then thetwo structure loops on OR are both stabilized by 7, so 7 = R, and there is aloxodromic element f mapping out of these boundary loops onto the other. ByC.20, every structure region stabilized by 7 is of the form f°(R). In particularR, = R2 = J. Then H, = H2 = J, and there is a loxodromic element f whichcommutes with J. This is equivalent to saying that the connector c has bothendpoints on the one part P, of signature (0, 2; v, v). Likewise there are twostructure regions, both kept invariant by R, with an R-invariant structure loopbetween them. We conclude that there is a connector c between the two specialpoints on P.

Observe that the argument above applies equally well if either H, or H. iselliptic cyclic.

D.12. From here on, we assume that J is elliptic, that H is finite, and that neitherH, nor H2 is elliptic cyclic.

Let H = cp*' (R), and let R be a structure region stabilized by H. Then R, R,and R2 are all stabilized by J. Let W be the structure loop on the boundary ofR separating it from R, and R2, and let = cp(W). Since cp is a similarity, l 'separates the elliptic fixed points of elements of R - 7 from those of both R, - 7and R2 - 7, even though .R lies between 1R, and A2. Let W. be the structure loopfor t7 separating A from Am. Notice that W. separates the elliptic fixed points ofR. - 7 from those of both R - 7, and R3-m - 7. It follows that 141 necessarilyintersects either 4P, or 1412. We assume that it intersects YD, in a non-trivialfashion; that is, we cannot deform these loops into loops which do not intersect,while keeping them invariant under 7, and staying in the complement of the fixedpoints of R, R, and R2.

Let 0 be the subgroup of C generated by R, R, and R2. The structure loop141, lies between the structure regions stabilized by R, and R, and, from eitherpoint of view, the outside disc bounded by it is precisely invariant under 7; hencewe can apply VII.C.2, to conclude that <R,, R> represents a sphere with fourspecial points, and that 141, projects to a simple loop on this surface. The pro-jection of 141, to this surface is a (power of a) simple loop that separates two ofthe special points from the other two. Similarly applying VHI.C.2 to <R,, R> and

X.D. Signatures 279

172, we see that 0 represents a sphere with five special points, where the projec-tions of IV, and W2 form a system of loops on Q(G)/G = S and separate it intothree building blocks, where two of these building blocks have two special pointsand one boundary component, and the third building block has one special pointand two boundary components. (The projection of W. is of course a power of asimple loop.)

Since W projects to a (power of a) simple loop on d/G, it also projects to a(power of a) simple loop on d/C, so it necessarily projects to a (power of a) simpleloop on S.

We deform IV, W,, and WZ so that they remain simple .7-invariant loops, butnow they have only simple crossings.

In Q(C), consider the set of all 0-translates of IV, think of these as heavy loops;also consider the translates of both W, and W2 as being light. The heavy curvesform a set of simple disjoint loops; so do the light curves.

Consider the set of intersections of a heavy curve and a light curve. This setof intersections divides each of these loops into segments. If x' and y' are twoadjacent points of intersection on the heavy curve, then they divide the heavycurve and the light curve into two segments each. It is easy to see that one of thesegments of the heavy curve contains no translates, under the stabilizer of theheavy curve, of either x' or y'.

W has non-trivial intersections with W, . These intersections divide each ofthese loops into segments. It is easy to find a pair of division points x' and y',where x' and y' are adjacent points of intersection on both W and W,; that is,there is an arc A of IP between x' and y', which contains no other points ofintersection with W, , and there is an arc B of W, between x' and y', which containsno other point of intersection with iI

Of course, there may be points of intersection with other light curves on A;we can find two such points of intersection, call them x and y, so that x and yare adjacent on A, and so that the same light curve W passes through both xand y. Let C be the arc of A between x and y, and let D be the arc of W betweenx and y, where D contains no Stab(W)-translates of x or y (see Fig. X.D.6).

Let 17' be the loop formed by traversing C from x to y, then D from y backto x. Observe that, except for traversing D, 9' has no points of intersection withany light loop (Of course there might be points of intersection with heavy loopsalong D). Since there are no translates of x or y on D, and there are no points ofintersection of C with any light curve, 9' projects to a simple loop. Let 9 be thesimple loop obtained from 17' by deforming it near x and y and along D, so thatit now runs parallel to D; that is, D has no points of intersection with any lightcurve.

Let v be the projection of P to S; then v is a loop which lifts to a loop, anddoes not intersect the projection of either W, or W2.

The first possibility is that v is homotopically trivial on °S. In this case, C andD together bound a disc that contains no elliptic fixed points and no limit points.Then we can deform 1V to follow D instead of C, and reduce the number of points


Fig. X.D.6

of intersection of iP with W, and W2 by two. Since there are necessarily pointsof intersection, we can assume that this does not happen.

Since v is simple, homotopically non-trivial, and does not cross the projectionof either W, or W2, it lies on one of the building blocks. Each of the buildingblocks is a sphere with three boundary components (either two special points,and one boundary loop, or one special point and two boundary loops). Hence vis freely homotopic either to a small loop about a special point, or the projectionof W,, or the projection of W2. But none of these are possible, for v is a simpleloop that lifts to a loop, and we know that the special points all have order atleast two, and that W, and W2 are precisely invariant under .1, which has orderat least two.

We have shown that there is no structure region A lying between A, and R2iit follows that there is a connector between the special points on K correspondingto the points x, and x2. This completes the proof of the fact that we can modifythe homeomorphism between the parts of K and those of K so as to preserve theconnectors.

D.13. We saw in D.3 that the Schottky number can be defined as the differencebetween the genus of S and that of K. Since S and S are homeomorphic, and Kand k are homeomorphic, t = 1. Q

D.14. It is clear that not every theoretical signature occurs as the signature of aB-group; for example, there is no B-group whose signature consists of exactlytwo surfaces of signature (0, 0). A theoretical signature a = (K, t) for which thereis a B-group with signature a, is called a real signature; in this case, we say thatthe B-group realizes the signature.

X.D. Signatures 281

D.15. Theorem. A real signature a = (K, t) satisfies the following.(A) If there is a part P with signature (0,0), then P is the only part of K.(B) There is no part with signature (0, 1; v).(C) There is no part with signature (0, 2; v1, v2), V1 # v2.(D) If P is a part with signature (0, 2; or,, oc), then there is no connector having

an endpoint on P.(E) If P is a part with signature (0, 2; v, v), v < oo, and c is a connector with one

endpoint on P, then the other endpoint of c is also on P.(F) If P, and P2 are distinct parts, each with signature (0, 3; 2, 2, 00), then there

is no o0-connector between P, and P2.

D.16. Proof of (A). Suppose there is a part P with signature (0, 0). Let Y be thepreimage of Re(P) in S; then Y is bounded entirely by 1-dividers, and Y is planar.Let w1, ..., w,, be the 1-dividers on Y. Suppose that these loops are nothomologically independent on S; then some subcollection of them divides S, sothere is at least one of them, assume it is w1, which appears only once on Y.

Since Y is planar, the product of all the boundary loops of Y, properly oriented,is freely homotopic to the identity. Hence w1 is freely homotopic to a product ofother dividers, contradicting the minimality of the number of dividers. Weconclude that the loops w1, ..., wp are homologically independent on S; henceeach one appears exactly twice on OY. The closure of Y, with these boundaryloops pairwise identified, is a subsurface of S. It is also a connected closed surface;hence it is S. We conclude that P is the only part of K.

D.17. Proof of (B) and (C). There can be no part with signature (0, 1; v) or(0, 2; v1, v,), v, 0 v,, because there can be no basic group with these signatures.

D.18. Proof of (D). Suppose there is a part P of signature (0, 2; cc, ce). Then thecorresponding structure subgroup H is parabolic cyclic. Suppose there is aconnector c with at least one endpoint on P. This means that there is a structureloop W on the boundary of a structure region R, lying over the real part of P,where Stab(W) = Stab(R) is parabolic cyclic.

This can occur only is H is contained in some preliminary structure subgroupH'; by C.21, H is a structure subgroup of H', and by IX.H.5, this cannot occur.

D.19. Proof of (E). Suppose there is a part P of signature (0, 2; v, v), I < v < oo.Then the corresponding structure subgroup H is finite cyclic. By C.20, if there isone H-invariant structure loop on the boundary of the corresponding structureregion, then there are two such structure loops, and they are G-equivalent. Thecorresponding statement for connectors is (E).

D.20. Proof of (F). Suppose we had two (0, 3; 2, 2, 00) parts P, and P2 with ano0-connector between them. Then there would be adjacent structure regions R,and R2, where H. = Stab R. has signature (0, 3; 2, 2, cc), and there would be an


cc-structure loop W between them. Then R, and R2 would each have twocc-structure loops on their boundary, with all four structure loops stabilized bythe same parabolic subgroup. This contradicts C.24. El

D.21. A signature satisfying conditions (A) through (F) above is said to beadmissable. We will show in X.F that every admissable signature is real.

D.22. Let p: 9 - S be a regular covering of a finite Riemann surface S, whereis planar. By the planarity theorem, there is a minimal set {w...... wk} of simpledisjoint loops on S so that p: 9 -+ S is the highest regular covering for which theseloops, when raised to certain positive powers, lift to loops. Exactly as in D.2, wecan construct the theoretical signature associated to this covering by cutting Salong these loops, filling in discs or punctured discs, and adding v-connectorswhere appropriate, 2 < v < co. We can repeat theorem D.15 for this case, asfollows.

If D.15(A) were false, then every divider would be a 1-divider; after cuttingalong the non-dividing 1-dividers, we would be left with a surface of genus zerowith a divider on it. This last divider is freely homotopic to a product of thenon-dividing 1-dividers.

If we had a part with signature (0, 1; v), v > 2, then one of loops w wouldsimultaneously be freely homotopic to a product of 1-dividers, and have v as thesmallest positive power for which it lifts to a loop (see III.C.2-3).

If there were a part with signature (0, 2; v, , v2 ), V, A v2, then one of our dividersw would have both v, and v2 as the smallest power for which it lifts to a loop.

In our situation, there are no co-connectors, so parts (D) and (F) do not apply.Finally, the minimality of the set of loops {w1,..., wk} assures that there can

be no part with signature (0,2; v, v), where there is a connector connecting aspecial point of this part with a special point of some other part.

X.E. Decomposition

E.I. In this section, we decompose function groups using combination theorems.This will both give us information about the structure of these groups, and showthat every function group can be built up from certain basic groups using com-bination theorems. The basic groups here are the elementary groups, Fuchsianand quasifuchsian groups, and degenerate groups. In the use of the combinationtheorems, the amalgamated or conjugated subgroup is always either trivial orelliptic or parabolic cyclic.

E.2. Let (G, d) be a function group, let S = d/G, and let {w1..... wk } be a completeset of dividers on S. Let w = wk be one of these dividers; let v be the integerassociated with iv, that is, w is a v-divider. As in B.1, let Y be a connectedcomponent of S - {w}, and let To be a panel lying over Y.

X.E. Decomposition 283

The set £ of all structure loops lying over w is a set of simple disjoint loops(as usual, if v = oc, we adjoin the appropriate parabolic fixed point to eachconnected component of p`(w)), which divides C into regions. Let T be theregion containing To and let Go = Stab(T) = Stab(T0).

Let W be a structure loop on dTo; then Stab(W) is either trivial, elliptic cyclic,or parabolic cyclic. Also, if Stab(W) is parabolic with fixed point x, then StabG(x)has rank 1; hence the results of X.B are applicable.

Note that if w is non-dividing, then Y is the only component of S - { w}, whileif w is dividing, it is one of the two components. Every structure loop on theboundary of To projects onto w in S. If w is non-dividing, then there are twoGo-equivalence classes of structure loops on 3T0, while if iv is dividing, there isonly one.

E3. Let W be a loop on OT., let B be the outside disc bounded by W, and letJ = Stab(W). Then B/J is a disc, or a disc with one point of finite ramification,or a punctured disc, which we consider to be a disc with one point of infiniteramification.

As in B.8, let To be the region To with all boundary loops and outside discsadjoined, except for parabolic fixed points. Then To = d(G0). If w is dividing,then Y has only one boundary component, and To /Ga is the surface Y with adisc sewn in along the boundary component; if v > 1, then there is exactly onespecial point of order v in this disc. If w is non-dividing, then Y has two boundarycomponents, and To /Go is Y with discs sewn in along these two boundarycomponents; if v > 1, then there is exactly one special point of order v is each ofthese discs.

We saw in B.8, that Go is a function group.

M. Proposition. If R c To is a G-structure region, then H = Stab(R) is aGo-structure subgroup.

Proof. By C.25, H is a maximal subgroup of G satisfying the following properties:4(H) is simply connected; (H, 4(H)) contains no accidental parabolic transforma-tions; and H contains every parabolic element of G whose fixed point lies onA(H). The first two properties are independent of G, and the third remains truefor any subgroup of G. Hence there is a Go-structure subgroup H' so that H c H'.Since H' c G, and H is maximal in G, H' = H.

E.5. Proposition. If H is a Go-structure subgroup, then there is a G-structure regionR e To, so that H = Stab(R); i.e., H is a G-structure subgroup.

Proof. H is a maximal subgroup of Go satisfying the properties mentioned above.As a subgroup of G, A(H) is still simply connected, and (H, 4(H)) still containsno accidental parabolic transformations. Suppose there is a parabolic elementj e G - Go whose fixed point x lies in 4 (H). Since j 0 Go, there is a structure loop


Won 0, To that separates To from j(T0). Since j is parabolic, and x E A(H) c aTo,x lies on W. By C.11, Stab(x) has rank 1, and every parabolic element of Stab(x)lies in Stab(W). Hence j e Go. Since His a Go-structure subgroup, and x C- A (Go),jEH.

We next show that H is a maximal subgroup of G with these properties. If His trivial, then every Go-structure subgroup is trivial (C.23); in particular thestabilizers of the G-structure regions contained in To are trivial. Hence, everyG-structure subgroup is trivial.

Assume that H is non-trivial, and that H is not maximal; i.e., there is aG-structure subgroup K H. Let R be a structure region stabilized by K. SinceK 4 Go, R 4 To. Since every G-structure region is either contained in To, or isdisjoint from it, there is a structure loop W' on OR that separates R from To.Then H c Stab(W'). Hence H is either trivial, which we have already eliminated,or elliptic cyclic, or parabolic cyclic. Let W, be the G-structure loop on aTo thatseparates To from R. Since H stabilizes both sides of W, , H = Stab(W, ). Inparticular, H stabilizes the G-structure region which lies in To and has Wt on itsboundary. That is, there is a G-structure subgroup H', so that H C-_ H' c Go. SinceH is maximal in Go, H = H'. Hence H is maximal in G.

E.6. We now know that the Go-structure subgroups are precisely the stabilizersof the G-structure regions in To. Two of these structure regions are G-equivalentif and only if their stabilizers are G-conjugate. Since To is precisely invariant underGo in G, an element of G mapping one of these structure regions into another liesin Go. Hence, two Go-structure subgroups are Go-conjugate if and only if theyare G-conjugate.

E.7. In general, structure loops and structure regions are not uniquely deter-mined; this is the cause of the awkward phrasing below.

Proposition. The G-structure loops in the interior of To can serve as a complete setof Go-structure loops.

Proof. Let W,...., W be a complete list of non-conjugate preliminary structureloops for G in the interior of To. Let W,, ..., Wq be small loops about branchpoints in A(G0), where there is exactly one loop for each Go-equivalence class ofbranch points. If v < cc, and w is dividing, then one of these loops can be takento be a boundary loop of To; if v < oo, and w is non-dividing, then there areexactly two of these loops that we can take to be boundary loops of To.

We first need to show that <p(W,),...,p(Wk),p(W,),...,p(Wq)> = No, thedefining subgroup for the covering p: °A(G0) - 'So; that is, p: °A(G0) - 'So is thehighest regular covering for which p(W, ),..., p(Wq ), when raised to appropriatepowers, all lift to loops. Of course, we know that these loops, when raised tothese powers do lift to loops in this covering; the question we need to answer is:Does every loop that lifts to a loop do so as a consequence of the fact that theseparticular loops lift to loops?


Let V be any loop in A(Go) = T,. By B.9, we can deform V inside Ta so thatit lies in To; it suffices to assume that V c To. Pick a base point and add spurs;then V is homotopic in d to a product of preliminary structure loops and smallloops about elliptic fixed points. Hence, in do, V is homotopic to a product ofpreliminary structure loops lying in To, small loops about elliptic fixed points indo, including boundary loops, and perhaps some loops lying in outside discs. Theboundary loops, and the loops lying in outside discs are homotopically trivial indo, so the projection of V to Y is homotopic to a product of projections ofpreliminary structure loops in to, and projections of small loops about ellipticfixed points in fia. This shows that No is normally generated by the projectionsof the preliminary structure loops and the small loops about branch points.

It is clear that none of the w, are redundant for the definition of No, for ahomotopy among the projections of the W. and W, on 'So, would equally wellbe a homotopy among the projections of these loops on S.

We now know that the preliminary structure loops for G in to can serve as aset of preliminary structure loops for Go.

Next let W' be an oo-structure loop in the interior of To. Since W' is in theinterior of To, the structure regions, R, and R2. on either side of W' are also inTo. Hence Stab(R,) and Stab(R2) are both contained in a preliminary structuresubgroup H G, and Stab(W') is accidental in H. Since W' is in the interiorof To, W' is not G-equivalent to any boundary loop. Hence if say Stab(R2) iselementary, then there is a second cc-structure loop on OR2, and the structureregion R3, on the other side of this other structure loop, is also in To, and Stab(R3)is non-elementary. Hence there are limit points of Go on both sides of W', soStab(W') is accidental in Go.

If W e aTo is an oo-structure loop, then of course J = Stab(W) is accidentalin G, but not accidental in Go.

It remains to show that every accidental parabolic transformation in Go isGo-conjugate to an oo-structure loop in fia. If J is a cyclic subgroup of accidentalparabolic transformations in Go, then J is accidental in some preliminary struc-ture subgroup H of Go. Since the preliminary structure loops for G which lieinside T. are exactly the preliminary structure loops for Go, there is a preliminarystructure subgroup H' for G, where H' H. Since J is accidental in H, everyJ-invariant loop separates A(H). Hence J is accidental in any larger B-group; inparticular, J is accidental in H', from which it follows that J is accidental in G.Hence there is at least one J-invariant cc-structure loops in A. Since J stabilizesTo, the fixed point of J cannot lie outside To; hence there is at least one J-invariantstructure loop in to. Since the stabilizers of oc-structure loops on aTo are notaccidental in Go, this J-invariant structure loop does not lie on aTo. El

E.8. We are now in a position to compute the signature ao = (K,, to) of Go.If w is a non-dividing v-divider, v > 1, then the parts of Ko are precisely the

parts of K, and the connectors of Ko are the same as those of K, except that thereis no connector in Ko corresponding to tv. That is, on So, instead of w, there are


two discs, each containing one point of ramification of order v. Also in this case,the genus of So is one less than the genus of S, and the genus of Ko is one lessthan the genus of K; hence to = t.

If w is a non-dividing 1-divider, then there is no connector in K correspondingto w, and the genus of S differs from the genus of K by at least one. Also So differsfrom S in that there are two discs sewn in along the two boundary curves obtainedby cutting S along w. In this case, Ko = K, and to = t - 1.

If w is dividing, then it divides S into two subsurfaces, call them S, and S2.Let W be some connected component of p-' (w), and let T, and T2 be the tworegions, cut out by £, on either side of W, where p(Tm) = Sm. Let Qm = (Km, tm) bethe signature of G. = Stab(T,). Since every structure region in d is G-equivalentto a structure region in either T, or T2, the set of parts of K is the disjoint unionof the set of parts of K, and the set of parts of K2. Every divider other than wlies in either S, or S2, and these, together with w, form a complete set of dividerson S. Hence the connectors in K, and those in K2 are also connectors in K.

If w is a 1-divider, then K is the disjoint union of K, and K2. Since thedifference between the genus of S and that of K is the sum of the differences ofthe genera of dm/Gm and Km, t = t, + t2.

If w is a v-divider, v > 1, then in addition to the disjoint union of K, and K2,K also contains the connector c corresponding to w. Of course, one endpoint ofc lies on K,, while the other lies on K2. As above, t = t, + t2.

E.9. Theorem. Let W be a v-structure loop on dTo, let J = Stab(W), and let B bethe closed outside disc hounded by W. Then B is a (J,G0)-block. B is strong if anyof the following conditions are satisfied.

(i)v<oo.(ii) G is geometrically finite.(iii) Go is geometrically finite.(iv) Every non-elementary structure subgroup of Go is quasifuchsian.

Proof. We already know from B.6 that B is a block, and that it is strong if anyof conditions (i), (ii), or (iii) hold. We now assume that v = cc, and that everynon-elementary structure subgroup of Go is quasifuchsian. Let R be the structureregion in To bounded by W, and let H = Stab(R). If H is elementary, then Hcontains a half-turn h, where h 0 J but hJh-' = J, so int(B U h(B)) is a pair ofdisjoint open discs that are precisely invariant under J in Go.

If H is non-elementary, then it is quasifuchsian, so there is a secondarycomponent .4'(H) on which it acts. Since the outside disc B is contained in d(H),d'(H) U B = 0. Also, since H is quasifuchsian, there is a precisely invariant discB' c d'(H). Then i U k is a pair of precisely invariant discs for J. El

E.10. We now finish our analysis in the case that w is non-dividing, and thenreturn to the case that w is dividing in E.15.


Proposition. Assume that w is non-dividing. Then every Go-structure subgroupis a G-structure subgroup, and every G-structure subgroup is G-conjugate to aGo-structure subgroup.

Proof. The first statement is just E.5. The second statement follows from E.4,together with the observation that since w is non-dividing, every building blockon S has a lifting to a structure region contained in T. E3

E.11. Since w is non-dividing, there are two structure loops W, and W2 on 8T,where W, and W2 are not conjugate in Go. Let B. be the outside disc boundedby W., and let J. = Stab(Wm) = Stab(Bm).

Every element of Go maps B, onto some outside disc bounded by a translateof W,, hence D2) is precisely invariant under (J,,J2). In fact, the closed discsare precisely invariant; this is obvious if v < oc, or if v = oo and the stabilizer ofthe parabolic fixed point on W, and on W2 is cyclic. The only other possibilityfor the stabilizer is for it to be infinite dihedral; but this occurs only when wis a dividing loop, and one of the building blocks cut out by w is a surface ofgenus zero, with exactly two special points on it, both or order 2. Since w isnon-dividing, this case does not occur. We have shown that (B,, B2) is preciselyinvariant under (J,,J2) in Go.

There is an element f e G - Go mapping W, onto W2. Since f is not in Go, fmaps the outside of W, onto the inside of W2. As we observed in VII.E.3, thesetwo statements are sufficient to guarantee that B, and B2 are jointly f-blocked.

In order to complete the hypotheses of VII.E.5, we note that since J, and J2are cyclic, they are surely geometrically finite. Also, the points of To are notGo-equivalent to any point of either B, or B2; hence the set A. of VII.E.5 is notempty.

We now know that the hypotheses of VII.E.5 are satisfied; hence the con-clusions hold for the combined group, <Go, f ). This includes conclusion (iii), forwe know that (B1,B2) is precisely invariant.

E.12. We next show that G = <Go, f ). Let g a G. If g tf Go, then g(T0) # To; letW, be the structure loop on 8To that separates To from g(T0), and let T, be thetranslate of To on the other side of IV,. Since W, is Go-equivalent to either W, orW2, there is a g, a <Go, f) which maps To onto T,; this element is either of theform g, = h of, or g, = h of -', for some h e Go. Now let WW be the structure loopon the boundary of T, that separates T, from g(T0), and let T2 be the translateof To immediately on the other side of WW. There is a g2 E <Go, f) which mapsT, onto T2; this element is of the form g2 = g, o h o J'o g or of the formg2 = g, o It of -' o g,'. Since a sequence of distinct translates of W, has diameterconverging to zero, after a finite number of steps, we reach g(T0). Since there isan element h in <Go, f) mapping To onto g(T0), h -' o g e Stab,(T0) = Go; henceg c- <Go,.f)


E.13. Theorem. In addition to the conclusions of VII.E.5, we also have the following.(xii) If d' # A is a component of G, then d' is G-equivalent to some non-

invariant component of Go; and(xiii) two structure subgroups of Go are conjugate in G if and only if they are

conjugate in Go.

Proof. Conclusion (xii) follows from conclusion (ix) of VII.E.5, together with theobservation that since J has at most one limit point on W, Q(G)/G and Q(Go)/Godiffer only in the projections of their invariant components. Conclusion (xiii)follows almost immediately from the fact that To is precisely invariant under Goin G, and, except for w, the projection of To covers all of d/G.

E.14. Before turning to the case that w is dividing, we make the remark that if witself lifts to a loop, so that J, = J2 = { 1), then G is the free product of Go andthe cyclic group <f>.

E.15. We turn now to the case that w is dividing. Let W be some structure looplying over w, let E be the set of G-translates of W, then E divides d into regions;Let T, and T2 be the regions on either side of W. Let J = Stab(W), and letGm = Stab(T,,,). Looking from the point of view of T,., W bounds a closed outsidedisc; call it B..

Proposition. Every structure subgroup of either G, or G2 is a G-structure subgroup.Every G-structure subgroup is G-conjugate to a structure subgroup of either G,or G2.

Proof. The first statement is just E.5. The second statement follows from E.4 andthe observation that every building block in S has a lifting to either T, or T2.

E.16. Since B, is an outside disc, it is a (J, Gm}block; in particular, Dm is preciselyinvariant under J in G.. If v < oo, then the entire closed disc is precisely invariant,but if v = oo, there is one case where this is not so; this occurs when the stabilizerof the fixed point x of J is not cyclic; i.e., the stabilizer is infinite dihedral.

If Stab(x) is not cyclic in say G,, then the structure region in T, with W onits boundary is infinite dihedral. It follows from C.24 that we cannot have twoadjacent G-structure regions whose stabilizers are both elementary and bothcontain the same parabolic cyclic subgroup. Hence if Stab,,(x) is not cyclic,Stabo2(x) is. It is easy to see that if StabRom(x) is cyclic, then the Gm-translates ofB. do not fill up all of B3_m; hence (h DZ) is a proper interactive pair.

We now know that the hypotheses of VII.C.2 hold; hence the conclusionsalso hold for (G,, G2>; this includes conclusion (iii), for W is precisely invariantunder J in either G, or G2.


E.17. We next show that G = <G1, G2 ). Let g e G. If g 4 G, then there is a structureloop W, on OT, separating T, from g(W), and there is a g, e G1 with g, (W) = WI;also of course g,(T2) lies on the other side of Wt. If g(W) lies on Og,(T2),then there is a g2eStab(g,(T2)) so that g2og,(W) = g(W); then g-1 og2og1eStab(W ). If g(W) is not on 3g, (T2 ), then there is a g2 a Stab(g, (T2 )) = g, G2g;'that maps W, onto the structure loop on the boundary of g, (T2) separating g t (T2 )from g(W). We continue in this fashion, and so generate a sequence of distincttranslates of W that a path from W to g(W) must cross. Since W is a block, thissequence is finite; that is, after a finite number of steps, we arrive at an elementhe <G,, G2 ), with h(W) = g(W). Hence G = <G,, G2 ).

E.18. Theorem. In addition to the conclusions of VII.C.2, we also have thefollowing.

(xii) If d' # A is a component of G, then d' is G-equivalent to a non-invariantcomponent of either G, or G2, and

(xiii) two G-structure subgroups H, and H, are G-conjugate if and only if thereis a structure subgroup H of either G, or G,, so that H, and H, are both G-conjugateto H.

Proof. Conclusion (xii) follows from VII.C.2(viii), together with the observationthat since J has at most one limit point on W, the invariant component of G isformed from the invariant components of G, and G2. Conclusion (xiii) followseasily from the fact that T, and T2 are disjoint, have disjoint projections to d/G,and, except for w, the projection of T, U T2 covers all of d/G. Q

E.19. Proposition. Let G be a function group whose signature (K, t) contains noconnectors. Then G is a free product, G = F * G, * ... * G, where F is a free groupof rank t, and G1. .... G is a complete list of non-conjugate structure subgroupsof G.

Proof. We inductively cut S = d/G along dividers as above. Each structure looplying over any of these dividers has trivial stabilizer. We observed in E.14 thatcutting along a non-dividing 1-divider splits G into a free product, where one ofthe factors is cyclic; there are exactly t such operations. We observed above thatcutting S along a dividing 1-divider splits G into a free product of two subgroups.We have shown that we can write G as a free product G = F * G1 s GR, whereF is free of rank t.

Each of the Gk can be viewed as the stabilizer of a region in d which coversa building block containing no structure loop in its interior; i.e., each G, is astructure subgroup of G; no two of these are conjugate, and there is one for eachconjugacy class of structure subgroups of G. 0E.20. Theorem. Let (G, d) be a function group.

(i) G is geometrically finite if and only if every structure subgroup of G isgeometrically finite.


(ii) If A' is a component of G, A' 96 A, then there is a unique quasifuchsianstructure subgroup H, so that A' = A'(H).

(iii) G is analytically finite.

Proof. We prove statements (i) and (ii) by induction on the number k of dividers;note that k = r2 + rO - 1 + t, where r2 is the number of connectors, and to isthe number of connected components of K; see D.2.

If k = 0, then A is simply connected. By IX.D.21, G is either elementary,quasifuchsian, degenerate, or G has accidental parabolic transformations. Sincethere are no oc-connectors, G has no accidental parabolic transformations. SinceG itself is the only structure subgroup of G, (i) is automatic. Elementary anddegenerate groups have only one component, so there can be another componentonly if G is quasifuchsian.

Now assume that (G, A) is given, where the number of dividers = k > 0, andthat our results hold for every function group with fewer than k dividers. Let wbe a divider on S = A/G.

E.21. We first take up the case that w is non-dividing. Let To be a region lyingover S - w, and let Go = Stab(T0). Note that there are exactly k - I dividerson So.

If G is geometrically finite, then by VII.E.5(x) Go is geometrically finite, soby induction, every Go-structure subgroup is geometrically finite. By E.10, theG-structure subgroups are conjugates of the Go structure subgroups, so everyG-structure subgroup is geometrically finite.

If every G-structure subgroup is geometrically finite, then, using E.10 again,every G0-structure subgroup is geometrically finite. By the induction hypothesis,Go is geometrically finite; then by VII.E.5(x), G is geometrically finite.

We turn now to statement (ii). Let A' be a non-invariant component of G.Then by E.I2(xii), A' is G-equivalent to a non-invariant component of Go; hencewe can assume that it is a non-invariant component of Go. By the inductionhypothesis, there is a quasifuchsian Go-structure subgroup H, so that A' = A'(H).Of course H is also a G-structure subgroup.

E.22. We now assume that w is dividing. Construct G, and G2 as in E. 15. Notethat both A,/G, and A2/G2 have fewer than k dividers on them.

If G is geometrically finite, then by VII.C.2(xi), G, and G2 are both geo-metrically finite. Then by the induction hypothesis, the G,-structure subgroupsand the G,-structure subgroups are all geometrically finite. Then by E.15, everyG-structure subgroup is geometrically finite.

If every G-structure subgroup is geometrically finite, then by E.15, everystructure subgroup of both G, and G2 is geometrically finite. Then by induction,G, and G2 are both geometrically finite. Then by VII.C.2(xi) G is geometricallyfinite.

X.F. Existence 291

For the second part, let d' be a non-invariant component of G. We knowfrom E.18.(xii), that d' is G-equivalent to a non-invariant component of eitherG, or G2; hence we can assume that d' is a non-invariant component of say G,.Then by induction, there is a G,-structure subgroup H so that d' = d'(H). SinceH is a G,-structure subgroup, it is a G-structure subgroup.

E.23. Statement (iii) follows almost at once from (ii). Since every quasifuchsiangroup is analytically finite (IX.F.10), every component of DIG is a finite markedRiemann surface. By C.27, there are only finitely many distinct conjugacy classesof structure subgroups; hence DIG has only finitely many components.

X.F. Existence

F.I. A function group in which every structure subgroup is either Fuchsian orelementary is called a Koebe group. We saw in E.20 that such a group is neces-sarily geometrically finite.

Theorem. Let a = (K, t) be an admissable signature. Then there is a Koebe groupwith signature a.

The main ingredients in the proof are the combination theorems; the proofproceeds via induction on the number of dividers on the underlying surface.Recall that the number of dividers is given by k = t2 + t0 - 1 + t, where t2 isthe number of connectors in the 2-complex K, to is the number of connectedcomponents of K, and t is the Schottky number.

F.2. If k = 0, then to = 1, t = 0, and K consists of exactly one marked surface P.Let p: d -+ P be the branched universal covering, where the branched universalcovering surface d is either the sphere, the plane, or the disc, and let G be thegroup of deck transformations. Then (G, A) is a function group with signature a(conditions B and C of D.15 are needed here to guarantee that the signature of(G, d) is indeed a).

F.3. Now assume that every admissable signature for which the number ofdividers is less than k can be realized by a Koebe group, and let a = (K, t) be anadmissable signature with k > 0 dividers.

We choose one of the dividers and delete it, as follows.(i) If t > 0, then let ao = (K, t - 1). By hypothesis, there is a Koebe group

(GO,A0) with signature ao.(ii) If t = 0, and to > 1, then let K, be a connected component of K, let

K2 = K - K and let am = (Km,O). Let G. be a Koebe group with signature am.This case is taken up starting with F.9.


(iii) If t = 0, ro = 1, and there is a v-connector c in K, v < co, where K - cis still connected, then let co = (K - c, 0), and let Go be a Koebe group withsignature a0.

(iv) If there is no v-connector as above, but there is a v-connector c, withv < co, then let K, and K2 be the two connected components of K - c; setam = (Kr,0), and let G. be a Koebe group with signature am. The argument forthis case starts with F.9.

(v) If every connector is an oo-connector, and there is a connector c that doesnot divide K, then let K0 = K - c, let a0 = (K0, 0), and let Go be a Koebe groupwith signature a0.

(vi) In the only remaining case, every connector is an oo-connector, and theyall divide K. Let c be one of these connectors, let K, and K2 be the two connectedcomponents of K - c, let am = (Km, 0), and let G. be a Koebe group withsignature am. The argument for this case starts with F.9.

F.4. We first take up cases (i), (iii), and (v). We are given the Koebe group (Go, d0)with signature co = (K0, to), and in cases (iii) and (v), we are given distinguishedpoints x, and x2, of order v, on K0, where these points are not endpoints of anyconnector in K0. Let So = d0/G0. Note that in case (v), t = 0, K is connected,and every divider is an oo-divider; hence Go is a B-group.

In case (i), choose some constrained fundamental set Do for Go, whereb0 = D0 fl do is connected, and let B, and B2 be disjoint closed circular discsin the interior of b0. Let Wm = OB.; note that J. = Stab(Bm) is trivial. It isimmediate that Bm is a strong (1, G0)-block, and that (B B2) is precisely invariantunder ({1},{1})inG0.

The remainder of this subsection is devoted to the construction of the anal-ogous strong blocks in cases (iii) and (v).

Since xm is not an endpoint of a connector in a0, there is a part Pm withxm a Re(Pm); it is not excluded that P, = P2. Since xm e Re(Pm), we can regard xmas lying on the corresponding brick Ym, and, since x, # x2, we can also regardx, and x2 as being distinct special points of order v on So.

Let U, and U2 be neighborhoods of x, and x2, respectively, on So, where U,and U2 have closures that are disjoint, connected, simply connected, and disjointfrom all dividers on So. Let A. be some connected component of the preimageof Um in do. We can assume without loss of generality that A, and E2 are bothopen circular discs. Let B. be the closure of ISm; if v = oo, B. includes theappropriate parabolic fixed point. Note that there is a maximal cyclic subgroupJ. of Go, where Jm has order v, so that B. fl o is precisely invariant under Jm.

We next observe that B, fl B2 = 0. Since their projections are disjoint, theonly other possibility is that they have a common parabolic fixed point; this canoccur only if there is a cyclic parabolic subgroup J of Go representing bothpunctures, x, and x2. Let H be the structure subgroup containing J; since Jrepresents two punctures on S = d0/G0, it also does so on d(H)/H; hence H is

X.F. Existence 293

not Fuchsian, and does not have signature (0, 3; 2, 2, cc). The only other pos-sibility is that H = J; i.e., the corresponding part has signature (0, 2; 0C, 0C). ByD.1 5(D), there can be no connector in K with an endpoint at either special pointof such a part.

The argument above shows that in fact B, is disjoint from every translate ofB2. We remark that B, is also disjoint from all its own translates. The only otherpossibility is that there is structure subgroup H containing J, = Stab(B, ), whereH is infinite dihedral. The connector c in K is an oc-connector with one endpointon the part P corresponding to H. By assumption, there are no v-connectorswith v < cc; hence there are no other connectors with one endpoint on P. Thiscontradicts the assumption that c is non-dividing. We have shown that (B,, B2)is precisely invariant under (J,, J2) in Go.

We still need to show that B = is strong, so we assume that v = cc,and let H be the structure subgroup containing J =.. We saw above thatH is non-elementary, hence Fuchsian. Then there is a precisely invariant discB' c A'(H). Since A'(H) is a non-invariant component of G, B' is disjoint fromB, and B U B' is precisely invariant under J in G; i.e., J is doubly cusped.

FS. In all three cases under consideration. B, and B2 are circular discs. Let f besome element of #t, where f maps the outside of B, onto the inside of B2. and fmaps the fixed points of J, onto those of J2. It follows that f conjugates J, ontoJ2; it might be that J, = J2, in which case f commutes with every element ofJ, = J2.

Note that the hypotheses of VIl.E.5 are all satisfied. That is, (B1,B2) isprecisely invariant under Q, J2)J2) in Go, and B, and B2 are jointly f-blocked. SinceB, and B2 are disjoint closed circular discs contained in 0, there must be pointsof Ao that are not covered by any translate of either B, or B2.

Let G = <Go, f>, let Do be a maximal constrained fundamental set for Go,where bo = Do fl Ao is connected, and dDo meets W, = aB, and W2 = dB2 in atmost finitely many points. Let 15 be 15, - (B, U B2), together with the requisitepoints of OB1, so that 15 is a connected component of the constrained fundamentalset D = (Do fl A) U (Do fl r)B, ), where A is the common exterior of B, and B2 (seeVII.E.5(viii)). Since Go is a function group, the elements of Go which identify thesides of bo generate Go. Hence the elements of G which identify the sides of 15(including f, which identifies W, with W2) generate G. It easily follows that if gis any element of G, then we can draw a path in Q(G) connecting 15 to g(6); i.e.,G has an invariant component A z) D. It follows from VlI.E.5(ix) that A/G isS - (U, U U2) where 8U, and OU2 are identified.

FA Our next goal is to show that (G, A) has signature (K, t). The loop W = 3B,has limit points of G on both sides (in particular, it separates the fixed pointsof f), and is precisely invariant under J = J, in G. Hence its projection is asimple loop, or power of a simple loop w, which is not null homotopic in A. In


cases (i) and (iii), it follows from the strong version of the planarity theorem thatw can serve as a divider on S = Q(G)/G.

In case (v), Go is a B-group. Since f maps the fixed point of J, onto that ofJ2, the limit set of G, which is the closure of the translates of 4(G0), is alsoconnected; that is, G is also a B-group. Since there are limit points on both sidesof W, , Stab(W,) is accidental, so its projection can serve as a divider on S. Weare now in a position to apply the results of the preceding section.

Let L- be the set of preimages of w in d, the invariant component of G. LetT = do be the region cut out by Z so that H = Stab(T) z) Go. By VII.E.5(ix),H=Go.

By E.10, the G-structure subgroups are precisely the conjugates of theGo-structure subgroups, and these are all Fuchsian or elementary; hence G is aKoebe group. Also from E.13(xiii), the set of conjugacy classes of G-structuresubgroups is precisely the set of conjugacy classes of Go-structure subgroups;hence the signature of G has exactly the same parts as does the signature of Go.

F.7. Proposition. The G-translates of the Go structure loops in do, together withthe G-translates of Wt can serve as a full set of structure loops for G.

Proof. We call the loops under consideration the tentative structure loops. Thestatement of the proposition involves several different statements; these are asfollows. First, every loop in °d is freely homotopic in °d to a product of tentativev-structure loops, v < rG, and small loops about elliptic fixed points in d. Second,if h e G is accidental parabolic, then h stabilizes some tentative oo-structure loop.Third, the projection of those loops stabilized by elliptic elements or the identity,is a minimal set of dividers. The fourth statement, which is clearly true, is thatthe stabilizer of a tentative oo-structure loop is accidental parabolic in G.

Since the set of tentative structure loops is a set of simple disjoint loops, whereeach of these loops contains at most one point not in °d, it is almost immediatethat any loop V in °d is freely homotopic to a product of loops, fl V,,,, where noVm crosses any tentative structure loop. Then each V. lies inside a translate ofa Go-structure region, call it Rm. Inside Rm, any loop is freely homotopic to aproduct of tentative v-structure loops, v < oo, and small loops about elliptic fixedpoints. Hence V is freely homotopic to such a product; this is equivalent to sayingthat p: `d - °S is the highest regular covering for the which the projections ofthe tentative v-structure loops, when raised to appropriate powers, together withsmall loops about special points, also raised to appropriate powers, lift to loops.

Suppose h is an accidental parabolic element of G, then by VII.E.5(iii), h isconjugate to some element ho of Go. Since h is accidental in G, the G-stabilizerof its fixed point has rank 1; hence the Go-stabilizer of the fixed point of ho hasrank 1. We note that while Stab(WI) is not accidental in Go, it is accidental in G,so there is nothing to prove if ho stabilizes a Go-translate of either W, or W2.

Now suppose that ho is not accidental in Go. Then there is a preliminarystructure subgroup H for Go containing ho. Then H is either a non-elementaryB-group or a rank I Euclidean group. In either case, there is a cusped region C

X.F. Existence 295

in A(H) for ho. Clearly, we can choose C sufficiently small so that it intersects noGo-translate of either W, or W2; hence C is also a cusped region for ho in G. Itfollows that hp is not accidental in G. We have shown that either ho is accidentalin G0, or ho stabilizes W,, or ho stabilizes W2.

If ho stabilizes either W, or W2, then It stabilizes the appropriate translateof W,. If ho is accidental in Go, then it stabilizes an oc-structure loop; then hstabilizes the appropriate translate of this structure loop.

The minimality of the set of tentative v-structure loops, v < cc, follows almostat once from the minimality of the set of Go-structure loops. If there were a freehomotopy in A among some set of tentative structure loops and small loopsabout special points, then this free homotopy would equally well hold in An,which contains A.

F.8. We can regard the parts of the signature of G as being the same as the partsof the signature of Go. Using this identification, we see from F.7 that the con-nectors in the signature of G are precisely the connectors in the signature of Go,together with one additional connector corresponding to W. We have shown thatK is the 2-complex in the signature of G.

We see from VII.E.5(ix) that S is obtained from So by cutting holes about thepoints x, and x2 (these are the discs B, and B2), and then gluing the boundariestogether. Hence S has two fewer special points than does So, and the genus of Sis one greater than the genus of So. In case (i), since the two 2-complexes are thesame, the Schottky number for G is one greater than the Schottky number forGo. In cases (iii) and (v), there are no 1-dividers on either S or So, so the Schottkynumber for both S and So is zero.

We have shown that G has signature a = (K, t).

F.9. We turn now to cases (ii), (iv), and (vi). We have two Koebe groups G, andG2. In case (ii), this is all we have; in cases (iv) and (vi), we also have picked outa distinguished point xm, of order v, on Km.

If we are in case (ii), choose some constrained fundamental set D. for Gm,where D. = D. n A. is connected, and choose a closed circular disc B. in theinterior of 6m. Normalize G, so that A, is the lower half-plane, and normalizeG2 so that,62 is the upper half-plane. Note that B. is a strong ({1}, Gm)-block.

In cases (iv) and (vi), there is a point xm in Km, where xm is the endpoint ofthe connector c. Since xm in not the endpoint of a connector in Km, there is apart P. c K. so that xm a Re(Pj). Then we can regard xm as being a special pointof order v on S. = dm/Gm. Exactly as in F.4, choose a neighborhood U. of xm,where the closure of U. is connected and simply connected, and does not intersectany divider on Sm. Let B. be the closure of a connected component of thepreimage of Um; we can assume without loss of generality that B. is a circulardisc, that 6m is precisely invariant under J. = Stab(Bm), and that B. fl S2(Jm) =B. fl Q(Gm); i.e., B. is a block. As in the preceding case, we normalize G. so thatA, is the lower half-plane, and 62 is the upper half-plane. We normalize G. furtherso that if J. is parabolic, then it is generated by z - z + 1, while if J. is elliptic,


it has its fixed points at ±i. With these normalizations, J, = J2 = J. Let W bethe common boundary of B, and B2; that is, W= P.

We next observe that B, is a strong block; there is nothing to show if J iselliptic; assume that it is parabolic. If the structure subgroup H. containing J isFuchsian, then there is a precisely invariant disc in the secondary componentd'(Hm); since d'(Hm) is disjoint from J is doubly cusped. If Hm is elementary,then condition D.15(D) assures us that it is not cyclic. Hence H. contains ahalf-turn h which anticommutes with every element of J; in this case B. U h(Bm)is a precisely invariant pair of discs.

F.10. We next verify the hypotheses of VlI.C.2. We already know that B. is a(J,Gm)-block; we still need to show that J e G J 96 G2, and that (B1,B2) is aproper interactive pair.

The first observation is that D.1 5(A) eliminates the possibility that J = G. istrivial. If J = Gm is parabolic cyclic, then the corresponding part Pm has signature(0, 2; oo, oo), and there is a connector with an endpoint on one of the special pointsxm of Pm; this possibility is eliminated by D.15(D). Similarly, D.15(E) asserts thatif J = G, is elliptic cyclic, then the connector c has both its endpoints on thecorresponding part, contradicting the assumption that c divides K. We haveshown that J * G, , and J # G2.

We saw in VII.C.2 that (B1,B2) is an interactive pair, and that it is properas soon as either D, fl B2 96 0 or D2 fl B, 0 0. Since the projection of Bm toQ(Gm)/Gm has non-empty exterior, both of these sets are not empty.

We now know that the hypotheses of VII.C.2 are satisfied, so we can use theconclusions. Let D. be a maximal constrained fundamental set for Gm, where aDmintersects W in at most finitely many points, and D, fl W = D2 fl W. Let 6m =Dm fl dm. Then the identifications of the sides of D. generate Gm. It is clear thatwe can choose Dm so that a generator of J identifies two of the sides of 6m, andthese sides do not lie entirely in Bm. Then 6 = (6, fl B2) U (62 fl B,) is a com-ponent of the constrained fundamental set (D, fl B2) U (D2 fl B,), and it is clearthat the identifications of the sides of ,6 generate G. If g is any element of G, thenwe can write it as a word in these generators, and so find a path in Q(G) con-necting 15 to g(6); this shows that G has an invariant component A.

It follows from VII.C.2(viii) that A(G)/G can be obtained from S, = d(G, )/G,and S2 = d(G2)/G2 by cutting out the discs U, and U2, and identifying theirboundaries. This shows that A/G is a finite Riemann surface. We have shownthat G is a function group.

F.11. Proposition. W is precisely invariant under J in either G, or G2.

Proof. It is clear that W is precisely invariant under J in both G, and G2 if J iseither trivial or elliptic cyclic. If J is parabolic, then W need not be preciselyinvariant under J in Gm, but if it is not, then the structure subgroup of G.containing J is infinite dihedral. By D.15(F), if the G,-structure subgroup con-

X.F. Existence 297

taining J is infinite dihedral, then the G2-structure subgroup containing J is not.Hence W is precisely invariant under the identity in either G, or G2.

F.12. We next show that G has limit points in both the upper and lower halfplanes. First assume that there is an element g, a G, so that g, (B,) fl B, = 0.Then g1(B1)c2. For any 92eG2 - J, 92og1(B1) c 92(A2) c A1. Henceg2 o g, is loxodromic with one fixed point in the interior of B, , and the other in

Now suppose that there is no such element g, in G1. Since B, is a block, forevery g, in G, - J, if g, (B1) fl B, 0, then J is parabolic, and g, (B,) fl B, is theparabolic fixed point on both W and g1(W). Hence g, fixes the parabolic fixedpoint on W. Then since g, (B,) fl B, 0 for every g, a G, , every element of G1has a fixed point at the parabolic fixed point on W; so G, is Euclidean. SinceQ(G1)/G, has a parabolic puncture, G, is parabolic cyclic or infinite dihedral.Since G, 96 J, G, is infinite dihedral. Similarly, if there is no g2 E G2 so thatg2(B2) c A,, then G2 is also infinite dihedral. The only way this can occur is ifK, and K, each have only one part of signature (0, 3; 2, 2, x), and there is anac-connector between the two parts. This cannot occur by D.I5(F).

F.13. In cases (ii) and (iv), since there are limit points in both the upper and lowerhalf planes, W projects to a homotopically non-trivial simple closed curve, orpower of a simple closed curve, w on S = JIG, where w is not freely homotopicto a small loop about a special point. By the strong form of the planarity theorem,w can serve as a divider on S. In case (vi), since every divider in both K, and K2is an oo-divider, G, and G2 are both B-groups. Since the limit sets of G, and G2meet at oo, A(G), which is the closure of the union of all the G-translates ofA(G1) U 4(G2), is connected, so G is also a B-group. Since there are limit pointson both sides of W, Stab(W) is accidental, and the projection of W can serve asa divider on S. We conclude that w can serve as a divider; hence the results ofX.E. are applicable.

It follows from E.15 that the structure subgroups of G are just the G-conjugatesof the structure subgroups of G, and G2; then by E.18(xiii), the set of G-conjugacyclasses of structure subgroups is just the union of the set of conjugacy classes ofG1-structure subgroups and the set of conjugacy classes of G2-structure sub-groups. This shows that the parts of the signature of G are exactly the parts of K.

F.14. Proposition. The G-translates of W, together with the G-translates of theG,-structure loops in d,, and the G-translates of the G2-structure loops in d2, canserve as a full set of structure loops for G.

Proof. We call this set of loops the set of tentative structure loops. The statementof the proposition involves several different statements; these are as follows. First,every loop in °d is freely homotopic in °d to a product of tentative v-structureloops, v < oo, and small loops about elliptic fixed points in A. Second, if h e G is


accidental, then h stabilizes some tentative structure loop. Third, the set of theseloops stabilized by elliptic elements, or the identity, is minimal.

Since the set of tentative structure loops is a set of simple disjoint loops, eachcontaining at most one point not in d, any loop V in °d is freely homotopic toa product of loops, n V,, where each V. lies in °d, and no V. crosses any tentativestructure loop. Then each V. lies in a translate of some structure region of eitherG, or G2, call it Rm. Inside Rm, V. is freely homotopic to a product of loops onthe boundary of Rm, these are just the tentative structure loops on MRm, and smallloops about elliptic fixed points. The same result therefore holds for V.

If h e G is accidental parabolic, then by VII.C.2(iii), since W is preciselyinvariant under J in either G, or G2, h is conjugate to some element ho of eitherG, or G2; assume for simplicity that ho e G, . Since h is accidental in G, it doesnot lie in any rank 2 parabolic subgroup of G, so it does not lie in any suchsubgroup of G,.

Assume that ho is not accidental in G and assume that ho does not stabilizeany translate of W. Let H be the G,-structure subgroup containing ho. If H isFuchsian, then ho is doubly cusped in G,, where one of the cusped regions liesin d,, and the other lies in the secondary component of H. If H is elementary,then ha is doubly cusped in G1, where both cusped regions lie in A,. In eithercase, we can make the cusped regions smaller so that they do not intersect anystructure loop. Since ho does not stabilize any translate of W, these cusped regionsdo not intersect any translate of B,. Hence ho is not accidental in G. We haveshown that every accidental parabolic element of G stabilizes a tentative structureloop.

Suppose there is a free homotopy among the tentative dividers with finitemarking. Let p: U - S be the branched universal covering of S, branched to thesame orders as the covering p: A - S, and let T be the covering group. We canlift this free homotopy to a relation among the lifts of the tentative dividers, raisedto appropriate powers, in F. Since no tentative divider crosses W, and T is a freeproduct, amalgamated across a lift of W, of a lift of S, and a lift of S2, we canhave such a relation only if there is a relation among the tentative dividerson S raised to appropriate powers, or there is a relation among the tentativedividers on S2, raised to appropriate powers. There can be no such relation, forthe dividers on both S, and S2 are minimal. Q

F.15. It follows from the above that the connectors in the signature of G areexactly those in the signature of G and those in the signature of G2, togetherwith the one additional connector corresponding to W, which has its endpointsat x, and x2. We have shown that the 2-complex in the signature of G is just K.

We see from VII.C.2(viii) that the genus of S is exactly the sum of the generaof S, and S2.

In case (ii), the genus of K is exactly the sum of the genera of K, and K2.Since the Schottky numbers for both G, and G2 are zero, so is the Schottkynumber for G.

X.G. Similarities and Deformations 299

In cases (iv) and (vi), K 1, K 2 and K all have exactly one connected component,and Gl and G2 have no 1-dividers; in these cases also, the Schottky number forG is zero. This completes the proof that G has signature or = (K, t).

Finally, again by 8.18, the G-structure subgroups are precisely the G-conjugates of the G1-structure subgroups and the G2-structure subgroups; theseare all either Fuchsian or elementary. Hence, G is a Koebe group. p

F.16. Combining F.1 with the remarks in D.22, we have proven that every planarregular covering of a finite surface can be realized by a Kleinian group. Moreprecisely, we have shown the following.

Theorem. Let p: 3' - S be a regular covering of the finite Riemann surface S, where5 is planar. Let F be the group of deck transformations on 9. There is a Koebegroup G, with invariant component d, and there is a homeomorphism 4: 9 -'.4, sothat tI,,: F - G is an isomorphism.

X.G. Similarities and Deformations

G.I. This section is devoted to proving that a function group is essentiallydetermined by its signature. There is an obvious equivalence of signatures; wesay that two signatures a = (K, t) and it = (R, t) are the same, or equal, if t = I,and there is an orientation preserving homeomorphism between K and R, wherethe homeomorphism preserves special points, and their orders.

G.2. Theorem. Let (G, A) and (C, d) be function groups.(i) There is a type-preserving similarity between (G, d) and (t, d) if and only if

G and G have the same signature.(ii) If G and Zi are both geometrically finite, then c is a deformation of G if

and only if G and 0 have the same signature.

G.3. If there is a type-preserving similarity between G and t;, then by D.5, G andti have the same signature. If t; is a deformation of G, where the deformationmaps d onto d, then the restriction of the deformation to d is a type-preservingsimilarity, so G and t; have the same signature. If the deformation does not mapd onto d, then C has two invariant components. Combining IX.D.2 and IX.D.21,this can occur only if 6 is quasifuchsian. For a quasifuchsian group, the two basicsignatures, given by the two components, are the same; in this case again, G and0 have the same signature.

The proof of the converse will occupy the rest of this section; we prove aslightly stronger result, as follows.

Theorem. Let (G, d) and (d, d) be function groups of the same signature a = (K, t).Then


(i) There is a type-preserving similarity between (G, 4) and (2,, d), and(ii) If G and Cr are both geometrically finite, then this similarity is the restriction

of a deformation.

G.4. The induction steps in the proof of this theorem are essentially the same asthose in the preceding section. That is, we induct on k = r2 + ro - I + t, thenumber of dividers on S = 4/G.

G.5. If k = 0, then G is a B-group without accidental parabolic transformations.Since G and 6 have the same signature, there is an orientation preservinghomeomorphism >%r between the marked surfaces S and S = d/c. Let t': 4 - dbe a lift of Vii. It is clear that 0 is a similarity; we need to show that it istype-preserving. Let x be a special point of order v on S, and let w be a smallloop about x. If v < oo, then w determines a geometrically primitive ellipticelement of G, while if v = oc, w determines a primitive parabolic element of G.Since ' preserves special points and their orders, if v < oo, the lifting of 41i(w)determines a geometrically primitive elliptic element of c of the same order; ifv = oc, then by IX.A.5, (w) determines a primitive parabolic element of C. Hencethe induced isomorphism rp = 0,*: G -- C is type-preserving on the elliptic andparabolic elements representing special points.

Every elliptic element of G represents a special point; there are parabolicelements of G that do not represent special points only if G is rank 2 Euclidean,in which case cp(G) = t is also rank 2 Euclidean (no other geometric basis groupcontains a rank 2 Abelian subgroup). Since the parabolic subgroup of a Euclideangroup is the unique maximal Abelian subgroup, (p is type-preserving in this caseas well.

We now prove part (ii) in the case that k = 0. If G is elementary, then so isC; we saw in V.H.3-4 that every type-preserving similarity between such groupsis the restriction of a deformation.

If G is geometrically finite and non-elementary, then it is quasifuchsian;similarly for C. We saw in IX.F.6 that every quasifuchsian group is a deformationof a Fuchsian group; hence we can assume that G and 2', are both Fuchsian. SinceG and 6 have the same signature, there is a type-preserving similarity 1po betweend and J. Then by IX.F.12, there is a deformation 0 of G onto C so that ('Po)s = (p.Since '-' o oo commutes with every element of G, we can extend it to be adeformation tit of G onto itself that is the identity outside A (see IX.D.4). Then0 o 41, is the desired deformation.

G.6. We now assume that k > 0, and that our theorem is true for all functiongroups with fewer than k dividers. In fact, we assume somewhat more. Each partof K corresponds to an equivalence class of structure regions in the invariantcomponent of G, and hence corresponds to a conjugacy class of structure sub-groups of G, and also of 0. We know from D.5 that every type-preservingsimilarity maps structure subgroups onto structure subgroups; we assume that


our similarity tfr maps any given structure subgroup of G onto a structuresubgroup of t lying over the same part. Further, we assume that i preservesspecial points with their orders, and preserves connectors. More precisely, ifR is a G-structure region, stabilized by the structure subgroup H, then 13 =(p(H) = tiHOr' is a structure subgroup which stabilizes some structure regionR. We form the sets R+ =J(H) and R+ = d(13); it is clear that one can extendOI R to R+ so as to map it homeomorphically onto A+, simply by filling in discs,and mapping special points to corresponding special points. This defines ahomeomorphism t,&: R+/H -+ A+/17. These are parts of K; we require that thesebe the same part, and that ii be the identity on the special points of thispart.

These additional hypotheses are consistent with the zero induction step,where there is only one structure subgroup, one part, and no connectors.

Using the result of the preceding section, we can assume without loss ofgenerality that t; is a Koebe group. As in F.3, we choose a divider w on S, anda corresponding divider w on 9 = d/C = p(d) as follows.

(i) If t > 0, then there is a non-dividing 1-divider w on S, and there is anon-dividing 1-divider w on S'.

(ii) If t = 0, and to > 1, then K is not connected, and every 1-divider dividesS into two subsurfaces. Choose a 1-divider w that divides S into two subsurfacesS, and S2 as follows. There are some number of dividers on each of these surfaces;cut these subsurfaces along these dividers, add discs or punctured discs, asappropriate, and add connectors, as appropriate. Call the resulting marked2-complexes K, and K2, respectively. Choose w so that K, is connected. Thereneed not be a corresponding divider w on S; the appropriate modifications willbe made in G.13.

(iii) If t = 0, to = 1, and there is a v-connector c in K, v < oc, so that K - cis still connected, then choose w and w to be dividers lying over c.

(iv) If there is no v-connector as above, but there is a v-connector c, withv < oo, then choose w and w to lie over c.

(v) If every connector is an oo-connector, and there is a connector c that doesnot divide K, then let w and w be dividers lying over c.

(vi) In the only remaining case, every connector is an oo-connector, and theyall divide K. Let c be one of these connectors, and let w and w be dividers lyingover c.

As in the preceding section, while there are special considerations for eachcase, there are two general categories; the odd cases use the second combinationtheorem; the even cases use the first.

G.7. We start with cases (i), (iii), and (v). Let £, 1, be the set of structure loopslying over w, w, respectively, and let T, 7 be one of the regions of C cut out by£, 1, respectively. Let Go = Stab(T), do = Stab(`). By E.3-8, Go and do arefunction groups, with invariant components do and do, respectively, and Go anddo have the same signature (KO, to), where in case (i), Ko = K and to = t - 1,


while in cases (iii) and (v), to = 0, and the total number of connectors in Ko isn - 1. In case (v), since every divider is an oo-divider, Go and to are bothB-groups.

By assumption, there is a type-preserving similarity epo: do -+ do, where 0opreserves the equivalence between parts of Ko and conjugacy classes of structuresubgroups. Let qpo = (0o), be the induced isomorphism.

By E.10, every structure subgroup of Co is also a structure subgroup of C,hence t;o is also a Koebe group.

In cases (iii) and (v), So = d0/Go has two special points, x, and x2, corre-sponding to the endpoints of the connector c. Similarly, o = Jo/Co has twospecial points, z, and x2, corresponding to the endpoints of c. Our additionalinduction hypotheses guarantee that the projected homeomorphism >[i: So -+ Somaps xm to zm.

We also have the assumption that if Go is geometrically finite, then thesimilarity 0o can be extended to a deformation. We remark that even if Go is notgeometrically finite, we can still extend 0o to a bicontinuous injection of Q(GO)into Q(60) as follows. Let H ..., Hq be a complete list of inequivalent quasi-fuchsian structure subgroups of Go. Each H. has a secondary component dm;every non-invariant component of Go is Go-equivalent to one of these (E.20). ByD.5, R. = t1Io(Hm) is also a structure subgroup of CO; the image subgroups,are mutually non-conjugate (Fuchsian) structure subgroups of Co. By assump-tion, there is a type-preserving similarity between d(H,) and 4(17m); hence(IX.F.12) we can find a similarity Y'.: A'(Hm) -+ d'(Am), where (q/.), = tpolH,,,. Wenow define 00 on d'(Hm) to be equal to 'Ym, and extend *o to all of Q(G0) byinvariance; i.e., Rio o g = tpo(g) o 4o.

From here on we regard the similarity 0o as being defined on all of Q(G0). Itfollows from E.13 and VII.E.5(x) that G is geometrically finite if and only if Gois; if G is geometrically finite, then we regard qio as being defined on all of C. Wecan assume without loss of generality that t/io1 Q(G) is smooth. There is also theprojected (smooth) homeomorphism / : So -* So, which preserves the order ofspecial points.

There are two structure loops W, and W2 on aT, where W, and W2 are notGo-equivalent, but there is an f c G with f(W,) = W2. Set Stab(Wm) = Jm, and letB. be the closed outside disc bounded by Wm. As we saw in E.11, (B1,B2) isprecisely invariant under (J, J2) and B, and B2 are jointly f-blocked. We remarkthat B, and B2 are both strong in cases (i) and (iii); also in case (v) if Go isgeometrically finite. We also observe, as in E.12, that G = (Go, f >.

Our next goal is to choose analogous objects, 1P1, W2, 71, 72, $ A2, andf, for C,, and then alter the homeomorphism 0o, so that it maps the disc B. ontoAm, and so that 0. of = f o 00 on 3B, . In cases (i), (iii), and (v), we have alreadychosen the divider i7v. In cases (iii) and (v), we know that 4o maps distinguishedpoints to corresponding distinguished points; in particular, we can choose I',and l'2 so that cpo(Jm) = Stab(I1m) = 7m. In case (i), J. is trivial, we choose W,and YY2 so that they both lie over w, and so that they are not do-equivalent.

X.G Similarities and Deformations 303

G.B. We temporarily restrict our attention to case (i). Choose a constrainedfundamental set Do for Go, where Do fl do is connected, and Do contains both B,and B2 in its interior (it is almost immediate that one can do this; simply connectp(B,) to p(B2) by a spur; then choose a full dissection of So, where no element ofthe dissection intersects p(B, ), p(B2) or the spur; then let Do fl do be some liftingof the complement of this dissection). Let Do = i/i(DO). Note that Do fl Jo is aconstrained fundamental set for the action of do on 2'o. Also, if Go is geometricallyfinite, then since t(io is a deformation, Do is a constrained fundamental set for Co.

Our goal is to deform 4/o, and with it Do, so that 00 maps B,. onto D andpreserves the identification on the boundary. Once we have achieved this, we canuse the action of G, and VII.E.5, to extend t/io from D to A.

Since we are free to deform the boundary of Do in a small neighborhood ofitself, we can assume that £D meets D, and D2, and all their do-translates in onlyfinitely many points.

Since D, and D2 project to compact subsets of So, they meet the projectionof C3Do in only finitely many arcs of sides of B,; of course some of these arcs maybe single points. Using the fact that p(D,) and p(112) are disjoint closed discs, itis easy to construct a homeomorphism &: So - So, homotopic to the identity,that maps f3(ODo) onto a set that misses both p(17,) and p(D2). There is a liftinga: Jo - Jo of I that commutes with every element of do (III.D.5). Set a(z) = zfor all z outside do. By IX.D.4, a o r/io is again a type-preserving similarity ordeformation; replace t/io by a o t/io. Note that with the new i/io, there are translatesof D, and D2 lying in the interior of Do = t/io(Do); replace D, and D2 by thesetranslates, and let f be the element of t; that maps Wt, the boundary of the newD,, onto I'Y2, the boundary of the new D2.

We now have t/io(B,) and D, both inside Do, but they need not agree.It is clear that there is another homeomorphism, homotopic to the identity,A: So - So, so that A maps p ° t/io(Bm) onto p(17,,), and A is the identity on p(r?Do).Let fl be the lift of A that commutes with every element of 0; replace t/io by fJ o t/ 0.We now have that 00 maps B. onto D,,,. Since all the maps in question areorientation preserving homeomorphisms, it is easy to deform Rio near aB, so thaton3B1,i/io°f =f°41o

Let D be the intersection of Do with the common exterior of B, and B2, withW, adjoined. By VII.E.5(viii), D is a constrained fundamental set for G. Theset Do = t/io(D0) might not be a fundamental set for 43, for there could be anoninvariant component of C that is not in the image of t/io; extend Do to aconstrained fundamental set by including appropriate sets in noninvariant com-ponents. Let D be the intersection of this new Do with the common exteriorof D and D2, with lT', adjoined. We now have a map i/i = 0o 1 D, mapping Dhomeomorphically onto a subset of D, and preserving the identification of thesides.

G.9. We now pick up cases (iii) and (v). We start with a constrained fundamentalset Do for Go, where Do is maximal with respect to both B, and B2; in fact, we


assume that aDo fl B. consists of two arcs meeting at the fixed point of Jm. B.projects to a disc with a distinguished point xm E So = do/Go (in case (v), xm is apuncture). We know that J. stabilizes an outside disc s,,, for do, where B. projectsto a neighborhood of xm on K. There is also an element f mapping aR, onto aD2.As in the preceding case, our goal is to deform 4o so that it maps B. onto 6m, oronto another structure loop with the same stabilizer, preserving the identificationon the boundary.

In case (iii), since p(Bm) and p(Rm) can be regarded as being neighborhoodsof the same point xm, p( m) n p o Oo(Bm) contains a neighborhood of zm.

In case (v), the point xm is not there, but jo(Bm) is a closed disc which isprecisely invariant under Jm, and Jm is accidental parabolic in 0, but not in Co.Since Co is a Koebe group, and J. is contained in a Fuchsian structure subgroupof Co, there is a precisely invariant cusped region for .7m in a non-invariantcomponent. Since there can be at most two disjoint precisely invariant cuspedregions, any two such regions contained in d must intersect; in particular,oo(Bm) n am # 0.

In both cases (iii) and (v), it is easy to find a homeomorphism I of So ontoitself, where a is the identity outside some connected and simply connectedneighborhood of X90 U p(t12) U p o 00(B,) U p o 00(BZ ); 6 maps p o tfio(B,) ontop($, ); and & maps fi o rfio(B2) onto p(B2 ). As above, let a be the lift to J of &, wherea commutes with every element of CO; seta equal to the identity outside J.Replace qo by a o 'o; after this replacement, 0o maps B. onto some Go-translateof Bm, which we now call $m; note that the new ,9m is precisely invariant underthe old J. (this case can actually occur; for example, in the case that J, is ahalf-turn in an even dihedral group). We also redefine f to map the outside ofthe new D, onto the inside of the new $2.

Finally, we deform 4o in a neighborhood of Bt so that after this deformation,/oofIaB, =.fo-/oI3Bi.

G.10. As in VII.E.5, let D be the intersection of Do with the common exterior ofB, and B2, with appropriate points of 3B, adjoined, so that D is a constrainedfundamental set for G. As in G.8, i/io(D) need not be a fundamental set for C; theremight be a noninvariant component not in the image of 0o. Since o maps B.onto Dm, preserving identifications, by VII.E.5(viii), ir(D) is a fundamental set forthe action of 6 on J. Hence we can extend i/ro(D) to a constrained fundamentalset b for t.

G.11. We now start the definition of our similarity or deformation i/i by setting0 I D = Rio I D (we have already done this in case (i)). As we observed in VII.D.6,there is an obvious isomorphism cp = 0,: G - , given by cp(g) = rGo o g o 0o t forg e Go, and tp(f) = .ff

We next define 41 on Q(G). If z E D, and g E G, set qi o g(z) _ cp(g) o t,& (z); thisdefines 0 on °Q(G). If z is an elliptic fixed point in Q(G), then z is G-equivalent


to some point on D. For an elliptic fixed point z on OD, set t/i(z) = tyo(z); thenfor a G-equivalent point g(z), set tli o g(z) = cp(g) o 0 (z).

Proposition. 41 I A is a type-preserving similarity.

Proof. Ifs is a side of Do fl Ao, and g mapping s onto s' is the corresponding sidepairing transformation, then for every z on s, t/i o g(z) = cp(g) o tfr(z); this followsat once from the fact that tyo is a similarity. Also, we have defined 0 so that forze W1, 0 -f(z) = f o t/i(z). This shows that 0 is continuous near D in A. It is easyto see that if i/i were not continuous at some point of A, then it would not becontinuous at a corresponding point of b fl A. Hence t/i 1 A is continuous.

We next need the fact that tfi maps A bijectively onto d; this follows from thefollowing set of facts: cp is an isomorphism; ty is injective on D, and maps D n donto D n d; and D fl d is a fundamental set for the action of 6 on J.

Since t/i is continuous and bijective, it is a homeomorphism of d onto J.Since 0o is a type-preserving similarity, the isomorphism it induces is type-

preserving. Since every non-loxodromic element of G is conjugate in G to anelement of Go, qp is type-preserving. Q

G.12. Continuing with cases (i), (iii), and (v), we turn now to the second part ofour theorem, and assume that G is geometrically finite. Then by E.13, Go isgeometrically finite, so by the induction hypothesis, o is a deformation of Goonto Co. We set 0IA(Go) _ 4oIA(Go).

Since every G-structure subgroup is conjugate to a Go-structure subgroup,every non-invariant component of G is a G-translate of a non-invariant com-ponent of Go. Since 00 is already defined on the non-invariant components ofGo, and 4io maps these homeomorphically onto the non-invariant componentsof Co, t' is defined, and equal to 0o, on the non-invariant components of Go. Weextend the definition of i to the other non-invariant components of G byt/i o g = Mp(g) o 0. Thus defined, maps Q(G) homeomorphically onto Q(C).

We know that f and f are both loxodromic; let ty map the attracting, respec-tively, repelling, fixed point off onto the attracting, respectively, repelling, fixedpoint of f.. If g e G, and x e A(Go) U A(< f >), set t/i o g(x) = cp(g) o t/i(x). Note thatt/i maps the union of the translates of A(G0) U A(< f >) onto the union of thetranslates of A(6) U A(< f >). Since t/io maps A(Go) bijectively onto A(do), and ,is an isomorphism, 0 is bijective on this set.

Now assume that x is some point for which O(x) has not as yet been defined.By VII.E.5(vi), there is a sequence {gm} of elements of G, where g(W,) nestsabout x. Of course W, lies, except perhaps for one point, in A, so 41 maps thissequence of loops homeomorphically into d; in particular, the set of loops{tG o gm(W, )} has the same separation properties that the original set of loops has.Since D, = t/i(B,) is a block, dia(0 o gm(WW )) - 0. We conclude that i o gm(W, )nests about some point, 1i(x).


It is clear that the definition of O(x) does not depend on the particularsequence { gm }; for tW, is precisely invariant under 71 = cp(J, ). Also the definitionis invertible; we could have started with a point z in 4(C), where x is not atranslate of a limit point of Co, or a translate of a fixed point of f; again byVII.E.5(vi), there is a sequence of translates of W, which nests about z, and ty'of this nested sequence of loops is again a nested sequence of loops about somepoint x, where x is not a translate of a limit point of Go or of <f ). We haveshown that 0: C - C is bijective.

Since Q is dense in C, it suffices to show that tG is continuous from inside 0;that is, if {zm} is a sequence of points of 0, and zm - x, then O(zm) -+ q'(x). Since00 is a deformation, 0 is continuous from inside any one non-invariant com-ponent. Write zm = gm(ym) where y, a D. Since any sequence of distinct translatesof D have (spherical) diameter converging to zero, we can assume that there issome fixed y e D so that y, = y. We will also assume that y e .J.

We first take up the case that x is a limit point of Go. Write zm = hm(tm), wherehm e Go, and tm a Do. There are now three cases to consider: either every tm lies inD, or every tm lies in either B, or B2 (for convenience, we assume it is B,), andthe elements hm represent distinct left cosets of J, in Go, or every tm lies in eitherB, or B2 (again, we assume it is B,) and every hm represents the same coset of J,.

If every t, lies in D, then o hm(tm) = (p(hm) ° 1(tm) = (Vo)(hm) ° lGo(tm) _410 o hm(t,) = 'YO(zm). Since 0o is a deformation, iko(zm) -+ 00(x) = 4/(x)

If every t, lies in B, and the hm represent distinct left cosets, then the setshm(B,) are all distinct, so dia(hm(B, )) 0. Also, the sets c o h,(B,) = 4i0 o hm(B, )are all distinct, so they also have diameter tending to zero. The sets hm(B, )necessarily converge to x; by the continuity of 4o, the sets 00 ° hm(B,) convergeto to(x) = fi(x). Hence IL(zm) - O(x).

If the t, all lie in B, and the hm represent the same coset of J, in Go, then wecan assume without loss of generality that the hm all lie in J,. Since the hm are alldistinct, Jm is parabolic cyclic; in this case, since the fixed point of J, is the onlylimit point of Go in B, , x is the fixed point of J1. Choose a generator j for J,, andfind a subsequence so that hm = j am, am increasing.

We need to show that the points t/i(tm) are bounded away from fi(x), the fixedpoint of 7t. To this end, recall that since J, is parabolic, G and d are B-groups.Also, since G is geometrically finite, the structure subgroup H, containing J, iseither quasifuchsian (including Fuchsian), or elementary; we saw in E.1I thatsince the projection w of W, is non-dividing, H is non-elementary. Similarly, thestructure subgroup H2 JZ is quasifuchsian. Since f -' maps the outside of W2onto the inside of W we have two disjoint cusped regions for J, accounted for;one is in the secondary component A'(H,) and the other lies in f -' (A'(H2)). Bothof these cusped regions are disjoint from A. Pick an axis, call it A, that crossesthe axis of J,. Since A is simply connected, A divides A into two regions; theregion between A andj(A), when intersected with A is bounded away from x. Weknow that 0 is continuous on both A and j(A), and the same reasoning showsthat the region inside J, between the images of A andj(A) is bounded away from

X.G. Similantics and Deformations 307

i(x). It is no loss of generality to assume that Do was chosen so that Don An liesbetween A andj(A). Then all the points 0(tm) lie between O(A) and 0 o j(A); hencethey are bounded away from ty(x).

It is now clear that since xm is increasing, the points t/i(zm) = tp(j 3.n) o q (tm) _j aT o t/i(tm) - t/i(x). This concludes the case that x is a limit point of Ga. It is clearthat ty is also continuous at translates of these limit points.

Next suppose that zm -+ x, where x is the attracting fixed point of f. Thenthere are elements f. e <f ), and there are points xm in A, the common exteriorof B, and B21 so that fm(xm) = zm. We can assume without loss of generality thatfm = f'. Then t/!(xm) lies in A', the common exterior of 11, and L12, and fm(A)converges uniformly to the attracting fixed point off We argue similarly if x isthe repelling fixed point of f. It is clear how to use the interaction of G, C, andrG to obtain the analogous result if x is a fixed point of a conjugate of f.

If O(x) is defined by a nested sequence of translates of W1, then for every mthere is a k, depending on m, so that zm lies between g,(W1) and gk+, (W1). Thenf/(zm) lies between cp(gk)(W,) and (Mgk+1101) Since k -4 x as m -s cc, fi(zm) -+O(x). This completes the induction step for cases (i), (iii), and (v).

G.13. We turn now to cases (ii), (iv), and (vi); that is, we assume that w is dividing.We start with some special considerations for case (ii), where we may have toconstruct w.

There need not be a 1-divider w on S dividing it into two subsurfaces lyingover K1 and K2. Let {w...... ivk} be the set of all 1-dividers on S; they divide Sinto subsurfaces F ..... 4k+1, where Pm lies over the connected component K.of K. We chose w so that K, is connected; hence we can assume that R 1 = K 1.Order the {wm} so that w1. .... wj are the dividers on the boundary of 1P,. If j = 1,we are finished; that is, we have found a 1-divider on S that divides it into twosubsurfaces, one of which lies over K1. If j > 1, then construct a simple loop iv-on S separating it into two subsurfaces, where one of these subsurfaces is planar,contains all the boundary loops w...... dj, and has no special points on it. Choosew so that it is freely homotopic to a product of the boundary loops, properlyoriented; that is, w is freely homotopic to

(X.I)/lwmY'.

m=1

We now change the set of dividers: replace iv, by w. It follows from (X.1) that thenormal closures of 10 ., ... . wj } and { w, w2, ... , wj } are equal. Hence replacing w,by w in the set of all dividers again yields a set of dividers (the minimalitycondition is easy to check). This new divider w divides S into two subsurfaces,one of which lies over K1. Choose a structure loop 1 ' lying over w.

Since w is a dividing divider, there are two regions bounded by translates ofone on either side, where one of these regions, call it T1 lies over K 1, and the

other, call it 72, lies over K2. Let C. = Stab(9`m). Then 6m is a function groupwith invariant component Am, and signature (K,,O). Hence there is a type-


preserving similarity ci,,,: d,. Jm, with induced isomorphism (P m: Gm Cm. If Gis geometrically finite, then so is Gm; in this case, the similarity is the restrictionof a deformation, which we call by the same name. The remark in G.7 about theintermediate cases is applicable here also; in any case, we can extend Wm to mapQ(Gm) homeomorphically into Q(Cm).

From the point of view of 1Pm, or C,,,' bounds an outside disc; call itAs in the preceding cases, our first goal is to construct a constrained funda-

mental set D for G, and a homeomorphism 0, mapping D onto a constrainedfundamental set 6 for 6, so that, by using the identifications on the boundary,

induces an isomorphism cp: G - C.We next alter the similarity 0. as in G.B. Choose a constrained fundamental

set D. for Gm, where D. contains B. in its interior. Then find a homeomorphism1m mapping 9m = dm/vm = Pm(Jm) onto itself, where &m is homotopic to the iden-tity, so that not only lies in the interior of &m o Pm o 0m(Dm), but p'm(Am) =1m o P. o Y1m(Bm). Let xm be the lift of &m, where am commutes with every elementof Gm. Replace q/ by xm o so that the new 'Ym maps W onto some Cm-translateof W. Now replace 0. by ym o Wm, where ym a C. is chosen so that the new Wm

maps W onto W. We now have 0m(Bm) = $m; in particular 0,(W) = 2(W). Since

both mappings are orientation preserving homeomorphisms, it is easy to deformsay 0, in a neighborhood of W, so that, after this deformation, i/i, I W = J021 W.

As in VII.C.2, define D = (D2 (1 B,) U (D, fl B2); since Dm fl W = W, D is aconstrained fundamental set for G. We define the homeomorphism i/i on D by.P I(D,nB2)_q,,I(D, nB2), and sp I(D2nB,)_02I(D2f1B,). Since 41, and /2agree on W, is continuous at every interior point of D.

G.14. We now pick up with cases (iv) and (vi), we will return to case (ii) in G.15.Here the loop w lies over a connector c in K, this connector divides K into twocomponents Kt and K2. There is a corresponding divider w on S, lying over c,where iv divides S into two subsurfaces. Since there are no 1-dividers on 9, thesetwo subsurfaces necessarily lie over K, and K 2.

Choose some structure loop W lying over w, and let £ be the set of allstructure loops lying over w. £ divides C into regions; let T, and T2 be the tworegions having W on their boundary, where Tm lies over Km. Let G. = Stab(Tm),and let Bm be the closed outside disc bounded by W (outside with respect to Tm,of course). Then Bm is a (J, Gm)-block, where J is the stabilizer of B. in Gm. LetSm = dm/Gm, and let xm be the special point on Sm corresponding to the endpointof the connector c; that is, Am projects to a neighborhood of xm on Sm.

Using the results of E.15-18, we know that G. is a function group, withprimary invariant component A., and with signature (K., 0).

We also choose some structure loop VP c d, lying over w; let I be the set ofall structure loops lying over w, let Tm be the two regions lying on either side ofW, where Tm lies over Km, let Cm = Stab(Tm), and let D. be the closed outsidedisc bounded by I'. Then C. is also a function group with invariant componentJm, and signature (Km,O). Let 9m = J./Cm = Pm(Am), and let zm be the special


point on S. corresponding to the endpoint of c; that is, int(,gm) projects to aneighborhood of Note that P. is strong.

By assumption, there is a similarity 'Ym: Am Llm, where rpm = (II/m)* maps astructure subgroup lying over a part P of K onto a structure subgroup lying overthe same part P, and 'Ym preserves special points and connectors. In particular,the projection Wm: Sm Sm, maps xm to

As in G.7, we extend 0. to the non-invariant components of Q(Gr), so that'Ym is a homeomorphism from Q(Gr) onto a (m-invariant set of components ofhim, and the isomorphism cpm: G. - C. is well defined. Of course, if G is geo-metrically finite, then by E.18 and VII.C.2(xi), G. is also geometrically finite; inthis case, we can assume that Y'm is a deformation.

As in G.9, we deform tm so that it maps pm(Bm) onto pm($m); it is clear thatwe can do this, for o p(Bm) and pm(D&) are both neighborhoods of Xm. The liftof this deformed Wm is a new similarity 0m, which now maps B. onto Dm, orperhaps some C.-translate of $m, with the same stabilizer. If necessary, we findan appropriate gm in Cm, and replace 0m by gm o Wm so that, after this change,0,.(Bm) = Finally, we deform say 0, near B, so that 0,1 W = 021 W.

Let Dm be a constrained fundamental set for Gm, where D. is maximal withrespect to Bm, and where D, agrees with D2 on W. Set D = (D, fl B2) U (D2 fl B, ),and define tit on D by I (D, fl B2) = tP, (D, fl B2 ), and t` I (D2 fl B,) _ q12 1 (D2 fl B, ).Since 0,1 W = ty21 W, a is continuous on D.

By VII.C.2(vii), D is a constrained fundamental set for G. Set D'm = tm(Dm),and observe that we can extend D. to a constrained fundamental set for 6mby adjoining appropriate sets in some of the non-invariant components; if Gis geometrically finite, then Dm is a constrained fundamental set for Cm. Wehave arranged matters so that D', fl W = D2 fl W. hence, again by VII.C.2(vii),

= (D, fl D2) U 02 fl D, ) is a constrained fundamental set for C. At least on d,agrees with tr(D), and if G is geometrically finite, then b = t/i(D).

G.15. We now resume the discussion of all three cases: (ii), (iv) and (vi). We have0 defined on D. We also have isomorphisms (pm: Gm - "m. As observed in VII.A.5,these define an isomorphism cp: G -+ C, where cp I G. = epm.

For any point z in °Q(G), there is a g e G, so that g(z) a D; set bi(z) _(p(g-') o t/ o g(z). Every elliptic fixed point in Q(G) is G-equivalent to some pointof OD, either in B, or B2; if say g(z) E OD n B, , then set ti(z) = cp(g`) o > i2 o g(z).We now have a well defined map ql: Q(G) Q(C).

Proposition. t, 4 is a type-preserving similarity.

Proof. We first show that is continuous on all of D. If D' is a translate of Dabutting D in B then there is a g E G2 so that D' = g(D). For all points nearD fl D', bi(z) = tu2(z), hence 0 is continuous at these points; similarly with the sidesof D in X. Since 1(i is continuous in a neighborhood of D in Q, it is continuousin all of A.


Since s'(D fl A) =F) n J is a fundamental set for t;, and since rp is an isomor-phism, 0 maps d bijectively onto J. Since / is continuous and bijective, it is asimilarity.

fir, and 02 are both type-preserving, and every non-loxodromic element of Gis G-conjugate to an element of either G, or G2; hence i/i is type-preserving. 0

G.16. We now assume that G is geometrically finite. Every non-invariant com-ponent d' of t; is stabilized by a quasifuchsian a-structure subgroup 17. ByE.15, R is a-conjugate to a structure subgroup 17 of either 6, or 02; we assumewithout loss of generality that R is a t;,-structure subgroup. Then (P-' (fl) is aG,-structure subgroup, necessarily quasifuchsian, for G, is also geometricallyfinite, and so has no degenerate structure subgroups. Since 0, is a deformation,it maps the secondary component of cp`(17) homeomorphically onto the sec-ondary component d`' stabilized by 17. Since 0 and f agree on the non-invariantcomponents of G, , 3' is in the image of cli. We have shown that li is a homeomor-phism of Q(G) onto O(R).

If x is a limit point of Gm, set O(x) = IIrm(x); if g e G, and x e A(Gm), then seto g(x) = p(g) o O(x). If x is a limit point of G, and x is not a translate of a limit

point of either G, or G2, then there is a sequence {gm} of elements of G, so thatgm(W) nests about x (VII.C.2(vi)). The images cp(gm)o fl(W) = nestabout some point x. Set fi(x) = x.

As in G.12, in order to show that 0 is continuous, it suffices to consider asequence {zm} of points of d, with zm - x c- A(G). We write zm = gm(y) where y issome point of D. Since the spherical diameter of gm(D) converges to zero, we canassume that y lies on W.

We first take up the case that x is a limit point of G,; the case that x is atranslate of a limit point of G, will then follow, and the case that x is a translateof a limit point of G, is almost identical. Then there are points {tm} in D1, andthere are elements {hm} in G,, with hm(tm) = zm. There are three case to consider:either the points tm all lie in B2, or the points tm all lie in B, and the elements hmrepresent distinct left cosets of G, /J, or the points tm all lie in B, and the elementshm represent the same cosec of G,/J.

If the tm all lie in B21 then they all lie in D, so we can take all the tm to be equalto y, and we can take gm = h,, E G, . In this case the result follows at once fromthe continuity of qi,.

If the tm all lie in B, and the hm represent distinct left cosets, then the translateshm(B,) are all distinct, so their (spherical) diameter tends to zero. Then hm(B,)converges to the same point as does hm(y). The corresponding sets cp, (hm)(A, )are also all distinct, and they converge to the same point that gyp, (hm) o 0, (y) _0, o hm(y) converges to. Of course this point is just fr, (x) = fi(x).

If the tm all lie in B, and the hm all represent the same left coset, then we canassume without loss of generality that the coset is the trivial one; i.e., the hm alllie in J. In this case, J is parabolic cyclic, x is the fixed point of J, and G and 0are B-groups. We have already observed that if the structure subgroup of G,

X.H. Schottky Groups 311

containing J is elementary, then the structure subgroup of G2 containing J isnon-elementary; it is of course quasifuchsian, and the secondary componentcontains a cusped region for J. If the G,-structure subgroup containing J isnon-elementary, then it is quasifuchsian, and there is a cusped region for J in thesecondary component; if it is elementary, then G, contains a half-turn thatanticommutes with the elements of J. In any case, the double cusp at x can bechosen to lie entirely in non-invariant components. The same statement is equallytrue for the double cusp at >/i(x), the fixed point of T.

Choose an axis A in G that crosses the axis of J, and choose a generatorj for J. We further specify the choice for Dm, by requiring that Dm lie betweenA and j(A); in particular, D, lies between A and j(A). Observe that 0 is welldefined on A, and on j(A), including the endpoints, which are fixed points ofloxodromic transformations. Hence D, fl A is bounded away from x, and like-wise, its image under 0 is bounded away from 'i(x). Write zm = j wheretm lies in D, fl A. Going again to a subsequence, we can assume that am oc.

Since the points 0(t.) = 0,(tm) are bounded away from fi(x), tr(zm) = (p(j am)o1' m) - 0W.

If x is not a translate of a limit point of either G, or G2, then it is definedby a nested sequence of translates of W. The image of this nested sequence oftranslates of W is a nested sequence of translates of 1V; which by definition nestsabout 4'(x). This completes the proof that 0 is continuous.

G.17. Since 01A(Gm) = I/rmlA(Gm), 0 maps A(Gm) bijectively onto A((im). If x isnot a translate of a limit point of either G, or G2, then by E.18 and VII.C.2(vi),the image of x is not a translate of a limit point of either C, or V2. Using nestedsequences of translates of lR we can define 0 on the set of limit points whichare not translates of limit points of ?, or of V2. We have shown that 0 iscontinuous and maps C bijectively onto C. hence it is a deformation.

X.H. Schottky Groups

H.I. A classical Schottky group of rank n is defined by 2n disjoint circles, C,,C;, ...,C C., which bound a domain D,= Q', and by n elements g,..... g, C- M,where gm(Cm) = C and gm(D) fl D = 0. The group G = <g,..... is the clas-sical Schottky group.

A Schottky group, not necessarily classical, is defined the same way exceptthat we permit the C. and C to be simple closed curves, not necessarily circles,but we still require g...... g to be elements of M. We sometimes say that G isthe Schottky group on the generators g, , ... , g,,.

Each of the loops Cm, C bounds two discs; the disc containing D is calledthe inside, the other disc is the outside.


H.2. Proposition. Let G he a Schottky group on the generators g...... g,,. Then(i) G is a function group with no non-invariant components.(ii) G is free on then generators g,, ...,(iii) G is purely loxodromic.(iv) D is a fundamental domain for G.(v) G is geometrically finite.(vi) The signature of G is (K, n) where K is an unmarked sphere.

Proof. The proof is by induction on n, and uses combination theorems.For n = 1, C, and C, are disjoint, and g, (D) fl D = 0, so g, has a fixed point

inside C, and a fixed point inside C, Hence G is loxodromic cyclic, free (on theone generator), and Kleinian. Since G is elementary, it is a geometrically finitefunction group. Also, since g, maps the outside of C, onto the inside of C;, Dis a fundamental domain for G (we also note for future use that D U C, is afundamental set for G). We can obtain Q/G by folding together the sides of D;this yields a torus T. The projection of C, is a simple loop on T, necessarilyhomotopically non-trivial, which lifts to a loop. Hence p(C,) is a non-dividing1-divider on T. On a torus, there cannot be a second homotopically distinctsimple disjoint loop, so p(C,) is the only divider. Cut T along this loop, and fillin two discs to obtain the marked 2-complex, K, which is an unmarked sphere.

If n > 1, let G. = <g...... let b0 be the (open) region bounded by CC',__ C.-I, C,'.-,, and let Do = U0 U C, U U C,,-,. By the induction hypothesis,Do is a fundamental set for Go. Let B, be the closed outside disc bounded by C,,,and let B2 be the closed outside disc bounded by C. Since (B,, B2) is preciselyinvariant under ({ 1 }, { Q) in Go, and g maps the outside of B, onto the inside ofB2, the hypotheses of VI I.E.5 are satisfied. We also note that B, and B2 are bothstrong.

Conclusions (ii), (iii), (iv), and (v) follow at once from VII.E.5(i), (iii), (viii), and(x), respectively. Since D is connected and the side pairing transformations of Dgenerate G, G is a function group. Since Go has no noninvariant components,neither does G.

Fold together the sides of D to see that S = Q/G is a closed Riemann surfaceof genus n, and the loops C,, ..., C. project to n homologically independentsimple disjoint loops, which lift to loops. Call these w, = p(C, ), ... , w = p(C).

It is fairly straightforward that {w,,..., can serve as a set of dividers onS. To show that there are enough of them, observe that every loop in Q is freelyhomotopic to a product of loops, each of which lies in a translate of D, and everyloop in D is freely homotopic to a product of boundary loops. The minimalityfollows at once from the fact that w,..... w are homologically independent.

The dividers wm are all I-dividers. Cut S open along w ..., w and fill in 2ndisjoint discs to obtain an unmarked sphere. Hence the signature of G is (K, n),where K is an unmarked sphere.

H.3. We regard the trivial group as a Schottky group on no generators.

X.H. Schottky Groups 313

Theorem. Let G he a function group. The following statements are equivalent.(i) G is a Schottky group.(ii) The 2-complex K in the signature of G is an unmarked sphere.(iii) Every structure subgroup of G is trivial.

Proof. We already know that (i) implies (ii). The equivalence of (ii) and (iii) followsfrom the fact that there is a one-to-one correspondence between the parts of K,and the conjugacy classes of structure subgroups, where the structure subgroupis the branched universal covering group of the part.

If there is only one structure region, then G is trivial, and there is nothing toprove. Suppose there is more that one structure region. Let D be a structureregion in J. We saw in C.23, under exactly these circumstances, that everystructure region is G-equivalent to D, that every structure loop is a 1-structureloop, that there are an even number of structure loops on the boundary of D,and that these structure loops are pairwise identified by elements of G. Sinceevery structure region is G-equivalent to D, D is a fundamental domain for theaction of G on A; since A is invariant, the elements which pair the structure loopson aD generate G. We have shown that there is a region D, bounded by 2n disjointloops, C,, C;, ..., C,,, C,,, and there are elements (g,,,}, withg(D)f1D=Q,andG=<gt,...,g.>.

H.4. For our next application, we will need two standard facts from combina-torial group theory. The first is that every subgroup of a free group is free [51pg. 95], and the second is that if G is (isomorphic to) the fundamental group ofa closed orientable surface of genus p > 0, then G is not free (to prove this, notethat if G were free, then by passing to the commutator subgroup, it would be freeof rank 2p, and so the 2p standard generators A ... , BB would freely generateG [51 pg. I 10]; but these 2p generators satisfy the one standard relation).

A Kleinian group is locally analytically finite if for every component A of G,A/Stab(A) is a finite Riemann surface.

HS. Theorem. Let G he a free, locally analytically finite, purely loxodromicKleinian group, Then G is a Schottky group; in particular, G is finitely generated.

Proof. Let A be a component of G, and let Go = Stab(A). Since G is locallyanalytically finite, Go is a function group with invariant component A. Let (K, t)be the signature of Go, let P be some part of K, let R be a structure region lyingover P, and let H = Stab6a(R). Then H is the branched universal covering groupof the marked surface P. Since H has no elliptic elements, there are no specialpoints of finite order on P; since H contains no parabolics, there are no specialpoints of infinite order on P (IX.A.5). That is, P is a closed orientable surface,and H is isomorphic to the fundamental group of P. Since H, as a subgroup ofa free group, is free, P has genus 0; i.e., H is trivial. Also, by D.15(A), since onepart P of K is an unmarked sphere, P is the only part; i.e., K = P. Then by H.3,Go is a Schottky group, of finite rank.


By H.2(i), Go has only the one component A. Since A = Q(G0) is dense in C,and A is precisely invariant under Go in G, every element of G must map A ontoitself. Hence G = Go.

H.6. Corollary. Let G be a finitely generated, free, purely loxodromic Kleiniangroup. Then G is a Schottky group.

Proof. By Ahlfors' finiteness theorem, since G is finitely generated, it is ana-lytically finite; in particular, it is locally analytically finite.

X.I. Fuchsian Groups Revisited

U. Most of the results in this section deal with Fuchsian groups of the secondkind. We start however with a general fact about Fuchsian groups.

Proposition. Let F be a Fuchsian group acting on H2, and let °S = (H2 fl °Q(F))/F.A simple loop w on °S represents an elliptic element of F if and only if it is freelyhomotopic to a small loop about a special point of finite order.

Proof. We already know that the lift of a small loop about a special point of finiteorder determines an elliptic element of F; we now assume that w is a simple loopthat determines an elliptic element of F, and that w is not freely homotopic to asmall loop about a special point of °S.

We first reduce the problem to the case that F is finitely generated. If F is notfinitely generated, then °S is an infinite surface. It is easy to find a subsurfaceso c °S where n, (So) is finitely generated, w lies in So, and w is not freelyhomotopic to a boundary loop, or small loop about a puncture, of So. Let T bea component of p'(S0), and let FO = Stab(T). Since there is a homomorphismfrom n,(So) onto Fo, Fo is a finitely generated Fuchsian group. The lifting of wdetermines the same elliptic element in Fo as it does in F; since w is not freelyhomotopic to a boundary loop, or small loop about a puncture in So, it is alsonot freely homotopic to a boundary loop, or small loop about a puncture, in(H2 fl °Q(F0))/Fo. We conclude that it suffices to consider the case that F isfinitely generated.

Since F is finitely generated, w is not freely homotopic to a boundary loopof s = (H2 fl Q(F)/F, for those are all represented by hyperbolic and parabolicelements of F.

For a finitely generated Fuchsian group F, °S = (°Q(F) fl H2)/F is a finitemarked Riemann surface from which a finite number of disjoint closed discs havebeen removed. We form a new surface S by filling in each of these closed discs,but marking exactly one point in each disc with the symbol "co". Then thebranched universal covering group of S is a Fuchsian group P of the first kind.

X.I. Fuchsian Groups Revisited 315

There is an obvious homeomorphism 0: S - S, where 1i,, = tp: F - P is anisomorphism. It is clear that fi(w) has the same properties that w has; hence wecan assume that F is of the first kind, and that °S is a finite surface.

Cut S along w, fill in two discs, with one special point each, and attach aconnector between these two special points. Mark the connector with the orderof the elliptic element of F representing w. Since w is either non-dividing, ordivides S into two pieces, each of which either has positive genus, or has at leastthree special points on it, this procedure yields an admissable signature. Let (G, d)be a Koebe group with this signature. Since p: °d - °S is the highest regularcovering for which w, and certain small loops about special points, whenraised to certain powers, lift to loops, it is a covering that is higher thanq: (0-02 fl °Q(F)) - °S. Fill in the special points of °S2(F), and their preimages in dto obtain a regular covering r: A -+ 0-02 (see III.F.8). Since HZ is simply connected,r is a homeomorphism; i.e., the two coverings p: °d °S, and q: (H2 fl °Q(F)) - °Sare equivalent. Since an element of F is parabolic if and only if it represents aspecial point of order oo, and every such special point is represented by aparabolic element of G, the isomorphism induced by r is type-preserving. Thesignature of (F, H2) has no connectors, and the signature of (G, d) does; thiscontradicts D.5.

1.2. Let F be a Fuchsian group of the second kind. Then the limit set of F is anowhere dense subset of the circle at oo, hence, qua Kleinian group, F has exactlyone component, necessarily invariant. If F is finitely generated, then H2/F is atopologically finite surface, and Q/F is its double; hence, if F is finitely generated,then it is a function group.

Since A (F) is closed and nowhere dense in P, every connected component ofA(F) is a point. Hence the limit set of any structure subgroup of F is a point;then every structure subgroup of F is either finite or Euclidean.

It is an exercise to go through the list of finite and Euclidean groups, and seewhich ones preserve a disc. The only such groups are elliptic cyclic and paraboliccyclic.

1.3. Proposition. Let G be a finitely generated Fuchsian group of the second kind.Then the signature of G is (K, t), where t >_ 0, and either K is an unmarked sphere,or K is a disjoint union of surfaces of signature (0, 2; a, a), 0 < 2 500; in particular,K has no connectors.

Proof. Since every structure subgroup is cyclic, the only possible basic signaturesfor the parts of K are (0, 0), and (0, 2; a, a). By D. I 5(D-E), we know that the onlypossibility for a connector is for the two special points on a part of signature(0, 2; a, a) a < oo, to be connected. In this case, there is a subgroup H c F, which,as an elementary group, is the group 1 + 74. It is easy to see that this groupkeeps no disc invariant.


1.4. Proposition. Let G be a function group, with signature (K, t), where either K isan unmarked sphere, or every part of K has signature (0, 2; a, a), 2 S a S co, andK has no connectors. Then G is a deformation of a Fuchsian group of the secondkind.

Proof. Every structure subgroup of G is elementary, hence geometrically finite,so by E.20, G is geometrically finite. Then by G.2, G is a deformation of anyfunction group with the same signature. Fuchsian groups of the second kind withall these signatures were constructed in VIII.D.

X.J. Exercises

J.1. Let S be a non-orientable surface; let fi: U - S be its universal covering, andlet F be the group of deck transformations. Call an element a in n, (S) orientationpreserving if a determines an orientation preserving element off. Let p: 9 -- Sbe a regular covering of S, where 99 is planar. If w is a loop on S, where w isorientation reversing, and wk lifts to a loop on S', then k is even.

J.2. If S is a non-orientable surface, and p: 9 - S is a planar regular covering ofS, then there is a homotopically non-trivial simple loop w on S so that w to somepositive power lifts to a loop.

J.3. State and prove a version of the planarity theorem for non-orientablesurfaces.

J.4. State and prove a version of the planarity theorem for infinite surfaces.

JS. Let G be a function group, and let E be a connected component of 4(G).Show that Stab(E) is a B-group.

J.6. Let G be a function group, and let E be a connected component of 4(G).If Z is not a single point, then Stab(E) is a non-elementary B-group withA(Stab(E)) = E.

J.7. Let G be an analytically finite Kleinian group, and let E be a connectedcomponent of A(G). Then Stab(E) is analytically finite, and A(Stab(E)) = E.

J.8. Let G be a geometrically finite Kleinian group, and let A be a component ofG. Prove that Stab(d) is geometrically finite.

J.9. List all the signatures that can occur as the signature of an elementary groupwith two limit points.

X.J. Exercises 317

J.10. Construct a function group G with the following property. There are threedistinct structure loops; no two of them are G-equivalent; and they all have thesame stabilizer, a cyclic group of order 2.

J.11. Same as J.10, but the stabilizer has order three.

J.12. If there are three structure loops, all with the same stabilizer, and thestabilizer has order greater than three, then two of them are equivalent.

J.13. If (G, d) and (d, 2) are B-groups, and there is a type-preserving isomorphismcp: G - t?;, then there is a possibly orientation reversing similarity 0: .4 -.3, sothat (p.

J.14. If G and d are Schottky groups of the same rank, and q : G is anisomorphism, then there is a possibly orientation reversing deformation 1i of Gonto 0, where 0,, = gyp.

J.15. Construct a Kleinian group that is locally analytically finite but notanalytically finite.

J.16. Let G be a locally analytically finite, purely loxodromic Kleinian group,which is free on n generators. Show that G is a Schottky group of rank n.

J.17. Let S be a marked Riemann surface. Call a loop w on °S universallyhyperbolic if w determines a hyperbolic element of the (unbranched) universalcovering of °S (this is the branched universal covering where all the special pointshave marking "oo"). Every universally hyperbolic element then has a universalgeodesic, given by the projection of the axis, from the unbranched universalcovering. Find a finitely generated Fuchsian group F, and a loop w on °S =(1-02 fl °Q(F))/F with the following properties: w is universally hyperbolic; w is theuniversal geodesic; w is not simple; and w determines an elliptic element of F.

J.18. Construct a function group (G, d), containing an elliptic element with fixedpoints x and y, so that

(a)xeA,yeA;(b)xeA,yEA;(c) x c- A, Y E Q - d (hint: for this part, the order of the elliptic element can be

at most six);(d)xed, yeA;(e) xEd, yEQ - A;(f)XE0-A,yEQ-A.

J.19. Let p: S' - S be a regular covering of the closed Riemann surface S, whereS is planar. Then there is a Schottky group G, with topological projectionq: Q(G) - S, and there is a covering r: 9 - Q(G), so that p = q o r.


J.20. If H is a structure subgroup of the function group G, then H is a maximalsubgroup of G with a simply connected invariant component, and with noaccidental parabolic transformations.

X.K. Notes

A.3. [57]. A.4. The planarity theorem first appeared in [57]; there is also a proofby Swarup [89], and an extension to infinite surfaces by Kulkarni [48]; the proofgiven here is due to Gromov (unpublished).

The topological classification of function groups appeared in a series ofpapers [58], [65], [67], and [68], with a slightly different notation (translate"factor", as in factor subgroup, factor loop, etc., in those papers, to "structure"here). Existence and uniqueness, in the conformal category, for those Koebegroups whose signature contains no connectors was given by Koebe [41];existence and uniqueness in general, also in the conformal category, appears in[67]. Koebe groups are also discussed by Marden [55]. F.16. [58]; a generaliza-tion to infinite surfaces appears in [61]. H.5-6. [59]. Some of these facts havebeen generalized to higher dimension (Gusevskii [31]), and to more generalgroups of motions (Kulkarni [47]). 1.1. This is in response to a question ofM. Sheingorn. J.5-7. These were observed by C. McMullen [72].

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Special Symbols

An 53f^53Clau5°d 115dia I. [6diaE 10EN 53

P 530n54M 1AA 2

®n 53.Cn [.

Stab 2-3tr 2A21A 32E' 11°Q 15Q 23< ...> 19, 244

11

Index

accidental parabolic transformation 222Accola, R.D.M. 222Ahlfors, L.V. 216Ahlfors' finiteness theorem 28 L75 205 249

314algebraic convergence 92analytically finite (Riemann surface or

(disconnected) Riemann surface) 22- - Kleinian group 28angle width L86

are 28associated orbifold 128

- 3-manifold L28atom 204axis 105 LQ6

(of an accidental parabolic transformation)222

full 232

B-group 214Baumslag, G. 212bending 178, 181, 212Bers, L. 232block L41

- strong L41boundary arc L06- element 196- geodesic L46- half-space L06- loop 256branch point-see special pointbranched regular covering 49- universal covering 50building block L02

Chuckrow. V. 97 240circle at infinity L03component of a Kleinian group 24- invariant 84- primary 243 260

secondary 264- subgroup 98conformal 55conjugate point 3connector 272convex region 65- - of a Fuchsian group LQ6covering 41- equivalent 42- regular 42curve 28cusp 118

exceptional L42excluded L42

cusped region 112cutting an edge L25cutting and pasting 32cycle of edges 62

infinite 29

cycle transformation 62- - infinite 22

deck group 44defining subgroup 42deformation 36

of a Fuchsian group 2113degenerate group 217, 236derived group 198dihedral group 88- - infinite 92 95dilation 54Dirichlet region 21disc, imaginary 225- removed 22discontinuity 23- free LS

dissection 28

divider 262

double dihedral group 102. 177doubly cusped region 112

Index 325

edge 68elementary group 23elliptic group 92- modular group 133essentially finite (fundamental polyhedron)

121

Euclidean group 91extended Fuchsian group 195- quasifuchsian group 196extremal length 214

face 68fiber 41filling a region 174Ford region 32, 22free product L36Fuchsian model 217function group 84 216fractional reflection 2fundamental domain 29fundamental polyhedron 69fundamental set 32- - constrained L3- - maximal 138 1.60

geometric generator (of an elliptictransformation) 36

Gromov, M 250

half-turn 7 85height 53HNN extension 157horoball L 1 L6- extended 120horocycle 2horospherc 11.6hyperbolic space 54

index (of a covering) 48induced isomorphism 36 85infinite edge 22inside boundary (of a convex region) 65interactive pair 136- - proper 138interactive triple 138- - proper 160invariant ball 64inversion (see also reflection) 54isometric circle 9- sphere 21

jointly blocked discs L60

Jorgensen. T. 19, 97

killing a component L25kind, of a Fuchsian group 103-- of a Kleinian group 168Klein's combination theorem 134Koebe group 291

length of a normal form, amalgated freeproduct L36

- - -- HNN extension 157lift 41, 46limit point 21limit set 21locally free group 198loop 28-- simple 28

marked Riemann surface 26- 2-complex 271minimal rotation 36Mobius group 54multiplier of a transformation 5

nested set of axes 148of simple loops 149of spheres 56

nice neighborhood 15 24 49Nielsen isomorphism theorem 109 227

235non-elementary group 23normal form for an amalgamated free

product 136for an HNN extension 157for a loxodromic or elliptic

transformation 6- for a parabolic transformation 5

normalization L6

outside boundary (of a convex region) 65outside disc 243 252

pair of pants 142panel 35part, of a signature 272- imaginary 215

real 215path 28Picard group 82 247 212point of approximation 22 L22planar surface 254planarity theorem 251Poincare metric 54Poincare's polyhedron theorem 75polyhedron 68

326 Index

precisely embedded 1.45invariant 24 3136

presentation 85preliminary divider) 59

structure loop 259region 259

subgroup 264

projection 41

puncture 21

quasifuchsian group L92.217, 232

ramification number 49 84point-see special point

rank (of a Euclidean group) 9.1reflection 2.58Riemann surface (disconnected) 25rotation 54

Schottky group 82 1 , 168. 184.3ll- infinite 121number 271

- type group 82 169Schwarzian derivative 248side 29. 68

pairing transformation 29 69signature, admissable 282

basic 84- of a Kleinian group 271

real 280- theoretical 272similarity 85 216simple (element of a Fuchsian group) 105sliding 178, 181, 191

small loop 49spanning disc 142special point 26 49sphere at infinity 60spur 45stabilizer 23stereographic projection 4 59strip 223structure loop 243 262

region 243 262- subgroup 243 263system of loops 192

Teichmuller space }ftThurston, W., 25 134 135 120topologically finite (Riemann surface) 26trace 2translation 54triangle group 91 92, 95true axis L06- - of an accidental parabolic transformation

222type-preserving isomorphism 32- similarity 217

unit 102universal covering 42

vertex 30vertical 53

wildly embedded circle 202

Yamamoto, H. 216

Grundlehren der mathematischen WissenschaftenA Series of Comprehensive Studies in Mathematics

A Selection

194, Faith: Algebra: Rings, Modules, and Categories 1191. Faith: Algebra II, Ring Theory192 Mal'cev: Algebraic Systems193, P61ya/Szego: Problems and Theorems in Analysis 1194. Igusa: Theta Functions195. Berberian: Baer'-Rings196. Athreya/Ney: Branching Processes191. Benz: Vorlesungen uber Geometrie der Algebren198. Gaal: Linear Analysis and Representation Theory199. Nitsche: Vorlesungen uber Minimalflachen200. Dold: Lectures on Algebraic Topology201. Beck: Continuous Flows in the Plane202. Schmetterer: Introduction to Mathematical Statistics203. Schoeneberg: Elliptic Modular Functions204. Popov: Hyperstability of Control Systems205. Nikol'skil: Approximation of Functions of Several Variables and Imbedding

Theorems206. Andrd: Homologie des Algbbres Commutatives201 Donoghue: Monotone Matrix Functions and Analytic Continuation20$ Lacey: The Isometric Theory of Classical Banach Spaces204. Ringel: Map Color Theorem210. Gihman/Skorohod: The Theory of Stochastic Processes I21L Comfort/Negrepontis: The Theory of Ultrafilters212 Switzer: Algebraic Topology - Homotopy and Homology211 Schaefer: Banach Lattices and Positive Operators217. Stenstrdm: Rings of Quotients218. Gihman/Skorohod: The Theory of Stochastic Processes II219. Duvant/Lions: Inequalities in Mechanics and Physics220. Kiritlov: Elements of the Theory of Representations22L Mumford: Algebraic Geometry I: Complex Projective Varieties222. Lang: Introduction to Modular Forms223.. Bergh/Lofstrom: Interpolation Spaces. An Introduction224. Gilbarg/Trudinger: Elliptic Partial Differential Equations of Second Order225. Schutte: Proof Theory226. Karoubi: K-Theory. An Introduction222. Grauert/Remmert: Theorie der Steinschen Raume228. Segal/Kunze: Integrals and Operators229, Hasse: Number Theory230. Klingenberg: Lectures on Closed Geodesics231. Lang: Elliptic Curves: Diophantine Analysis232. Gihman/Skorohod: The Theory of Stochastic Processes III233. Stroock/Varadhan: Multidimensional Diffusion Processes234. Aigner: Combinatorial Theory235. Dynkin/Yushkevich: Controlled Markov Processes236. Grauert/Remmert: Theory of Stein Spaces231 Kothe: Topological Vector Spaces II238. Graham/McGehee: Essays in Commutative Harmonic Analysis239. Elliott: Probabilistic Number Theory I

240. Elliott: Probabilistic Number Theory 1]241. Rudin: Function Theory in the Unit Ball of C"242. Huppert/Blackbum: Finite Groups II243. Huppert/Blackbum: Finite Groups III244. Kubert/Lang: Modular Units245. Comfeld/Fomin/Sinai: Ergodic Theory246. Naimark/Stem: Theory of Group Representations247. Suzuki: Group Theory I248. Suzuki: Group Theory II249. Chung: Lectures from Markov Processes to Brownian Motion250. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations251. Chow/Hale: Methods of Bifurcation Theory252. Aubin: Nonlinear Analysis on Manifolds. Monge-Ampere Equations253. Dwork: Lectures on p-adic Differential Equations254. Freitag: Siegelsche Modulfunktionen255. Lang: Complex Multiplication256. Hormander: The Analysis of Linear Partial Differential Operators I257. Hormander: The Analysis of Linear Partial Differential Operators II258. Smoller: Shock Waves and Reaction-Diffusion Equations259. Duren: Univalent Functions260. Freidlin/Wentzell: Random Perturbations of Dynamical Systems261. Bosch/Guntzer/Remmert: Non Archimedian Analysis - A Systematic Approach to

Rigid Geometry262. Doob: Classical Potential Theory and Its Probabilistic Counterpart263. Krasnosel'skii/Zabrelko: Geometrical Methods of Nonlinear Analysis264. Aubin/Cellina: Differential Inclusions265. Grauert/Remmert: Coherent Analytic Sheaves266. de Rham: Differentiable Manifolds267. Arbarello/Comalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol. I268. Arbarello/Comalba/Griffiths/Hams: Geometry of Algebraic Curves, Vol. II269. Schapira: Microdifferential Systems in the Complex Domain270. Scharlau: Quadratic and Hermitian Forms271. Ellis: Entropy, Large Deviations, and Statistical Mechanics272. Elliott: Arithmetic Functions and Integer Products273. Nikol'skii: Treatise on the Shift Operator274. Hi rmander: The Analysis of Linear Partial Differential Operators III275. Hormander: The Analysis of Linear Partial Differential Operators IV276. Liggett: Interacting Particle Systems277. Fulton/Lang: Riemann-Roch Algebra278. Barr/Wells: Toposes, Triples and Theories279. Bishop/Bridges: Constructive Analysis280. Neukirch: Class Field Theory281. Chandrasekharan: Elliptic Functions282. Lelong, L. Gruman: Entire Functions of Several Complex Variables283. Kodaira: Complex Manifolds and Deformation of Complex Structures284. Finn: Equilibrium Capillary Surfaces285. Burago,V. A. Zalgaller: Geometric Inequalities286. Andrianov: Quadratic Forms and Hecke Operators287. Maskit: Kleinian Groups288. Jacod/Shiryaev: Limit Theorems for Stochastic Processes

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