Ergebnisse der Mathematik und ihrer Grenzgebiete

3. Foige

A Series of Modern Surveys in Mathematics

Editorial Board

S. Feferman, Stanford M. Gromov, Bures-sur-Yvette J. Jost, Leipzig J. Kollar, Princeton H.W. Lenstra, Jr., Berkeley P.-L. Lions, Paris M. Rapoport, Koln J.Tits, Paris D. B. Zagier, Bonn

Managing Editor R. Remmert, MUnster

Volume 40

Springer-Verlag Berlin Heidelberg GmbH

Luis Ribes Pavel Zalesskii

Profinite Groups

Springer

Luis Ribes School of Mathematics and Statistics Carleton University Ottawa, Ontario KlS ?B6, Canada e-mail: lribes@math.carleton.ca

Pavel Zalesskii Department of Mathematics U niversity of Brasilia 70910-900 Brasilia DF, BraziI e-mail: pz@mat.unb.br

Library of Congress Cataloging-in-Publication Data applied for

Die Deutsche Bibliothek - CIP-Einheitsaufnahme Ribes. Luis: Profin~e groups / Luis Ribes ; Pavel Zalesskii. - Berlin; Heidelberg ; New York; Barcelon ; Hong Kong ; London ; Milan ; Paris; Singapore ; Tokyo: Springer. 2000 (Ergebnisse der Mathematik und ihrer Grenzgebiete; Folge 3,Vol. 40) ISBN 978-3-642-08632-8 ISBN 978-3-662-04097-3 (eBook) DOI 10.1007/978-3-662-04097-3

Mathematics Subject Classification (1991):

20E18, 20J05, 12G05, 20E05, 20E06

ISSN 0071-1136

ISBN 978-3-642-08632-8

This work is subject to copyright. AII rights are reserved, whether the whole or part of the materi­al is concemed, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, '965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the Ger­man Copyright Law.

© Springer-Verlag Berlin Heidelberg 2000 Originally publisbed by Springer-Verlag Berlin Heidelberg New York in 2000 Softcover reprint of the hardcover 1 st edition 2000

Typesetting: Camera-ready copy produced by the authors' output file using a Springer T EX macro package Printed on acid-free paper SPIN 10479976 44/3143/LK - 5 43210

Para Marisa, Alfonso y David

Jboo,M,U.Ae U A.I&u1CY

Preface

The aim of this book is to serve both as an introduction to profinite groups and as a reference for specialists in some areas of the theory. In neither of these two aspects have we tried to be encyclopedic. After some necessary background, we thoroughly develop the basic properties of profinite groups and introduce the main tools of the subject in algebra, topology and homol­ogy. Later we concentrate on some topics that we present in detail, including recent developments in those areas.

Interest in profinite groups arose first in the study of the Galois groups of infinite Galois extensions of fields. Indeed, profinite groups are precisely Galois groups and many of the applications of profinite groups are related to number theory. Galois groups carry with them a natural topology, the Krull topology. Under this topology they are Hausdorff compact and totally dis­connected topological groups; these properties characterize profinite groups. Another important fact about profinite groups is that they are determined by their finite images under continuous homomorphisms: a profinite group is the inverse limit of its finite images. This explains the connection with abstract groups. If G is an infinite abstract group, one is interested in deducing prop­erties of G from corresponding properties of its finite homomorphic images. The kernels of all homomorphisms of G into finite groups form a fundamen­tal system of neighborhoods for a topology on G, and completion of G with respect to this topology gives a profinite group. In the last decades there has been an extensive literature on profinite groups and one of the aims of this book is to present some of these important results.

The first comprehensive exposition of the theory of profinite groups ap­peared in the book 'Cohomologie Galoisienne' by J-P. Serre in 1964. Its em­phasis is on cohomological properties and their applications to field theory and number theory. This deceptively slim volume contains a wealth of infor­mation, some of it not found elsewhere. We have learnt a great deal from Serre's book throughout the years and this, no doubt, is reflected in our exposition in the present book.

We describe briefly the contents of our book. The first three chapters deal with the basic tools and the main properties of profinite groups. In Chapter 1 we have collected information about inverse and direct limits and their alge­braic and topological properties, which is used throughout the book. Chapter

VIII Preface

2 contains a fairly detailed account of general profinite groups. The results are presented in the context of pro -C groups (inverse limits of groups in C), where C is a convenient class of finite groups, which includes the classes of profinite and pro-p groups as particular cases. The minimum we require of such a class C is that it should be a 'formation' (Le., closed under tak­ing quotients and finite sub direct products); but often we assume that C is a 'variety' (Le., closed under taking subgroups, quotients and finite direct prod­ucts). Although this approach requires the reader to become familiar with a little more terminology (but not much more than what is indicated above), this is compensated by being able to bring many related concepts and results together. Sometimes we assume throughout a chapter or a section that C satifies certain conditions; when that happens we indicate those assumptions in italics at the beginning of the chapter or section.

The main properties of free profinite (pro - C) groups are developed in Chapter 3. These includes several useful characterizations in terms of lifting maps it la Iwasawa and the study of the structure of open subgroups of free pro - C groups. Chapter 4 considers properties of particular profinite groups, including profinite abelian groups, Frobenius profinite groups and automor­phism groups of finitely generated profinite groups.

Chapters 5-7 deal with homological aspects of profinite groups. In Chap­ter 5, we consider modules over profinite rings, particularly complete group rings, and constructions involving them. Chapter 6 establishes the fundamen­tal results of homology and cohomology groups of profinite groups. Here we combine a computational approach with a conceptual one: on the one hand, we define homology and cohomology groups by means of standard resolu­tions, and on the other hand, we give a more abstract description, using the language of universal functors. Chapter 7 contains cohomological character­izations of projective profinite groups and the Tate characterization of free pro-p groups.

Chapter 8 considers closed normal subgroups of free profinite groups, and in particular, conditions under which such subgroups are free profinite. We also study similar properties for closed subnormal subgroups and accessible subgroups. This chapter includes Mel'nikov's theory of homogeneous groups, which gives a description of certain closed subgroups of free pro-C groups (other than pro-p).

Chapter 9 establishes the main properties of the basic 'free constructions' of profinite groups: free and amalgamated products and HNN-extensions. This is the beginning of the theory of profinite groups acting on 'profinite trees', which we shall develop in a subsequent book.

The last section of each chapter gives some of the history of the theory that has been developed, and indicates the names of the main contributors. These sections also include statements or references to results not treated in the main body of the chapters.

Preface IX

Throughout the text we have included a series of open questions that are also gathered at the end of book.

We thank Hendrik Lenstra Jr. for his suggestion that a book such as this should be written for the Ergebnisse Series. His contagious optimism and enthusiasm, and his interest in our ideas and projects have been very uplifting and helpful.

Several colleagues and friends have read parts of the book. We are spe­cially grateful to Zoe Chatzidakis, Juan Ramon Delgado, John Dixon, Otto Kegel and Wolfgang Herfort for their comments and corrections; the errors and misprints that may remain are attributable entirely to us. We are greatly indebted to Jean-Pierre Serre for sharing with us some of his ideas and for his help in Section 6.9.

Part of this book was written while one of us (LR) was on sabbatical at the UNED in Madrid at the invitation of Emilio Bujalance. The congenial mathematical atmosphere that our colleagues have created there was very conducive to our work. It is a pleasure to thank them for wonderful discussions (mathematical and otherwise) and for their friendship. The advice of Javier Perez regarding xy-pic was very useful and we thank him for the time he spent teaching us the tricks.

In the Summer of 1988 both authors participated in the program Research in Pairs of the Mathematisches Forschunginstitut in Oberwolfach while writ­ing this book; we thank the Mathematisches Forschunginstitut for the use of the excellent Library there and for the opportunity to work together and uninterrupted in such quiet and confortable quarters in the beautiful and relaxing Schwarzwald.

The first author gratefully acknowledges the support of the National Sci­ence and Engineering Research Council of Canada and the Direccion General de Investigacion y Desarrollo of Spain.

The second author thanks the Austrian Science Foundation and FUnda~ao de Apoio a Pesquisa do Distrito Federal (Brazil) for support.

Responsability for the writing of this book: L. Ribes has written most of the material in Chapters 1-8; the main exceptions are Section 4.5 and parts of Sections 4.4, 4.7, 5.6 and 8.3 which were written by P. Zalesskii; translation from Russian done by P. Zalesskii was important in the writing of Sections 8.5 and 8.10. Chapter 9 has been written by both authors. The editorial work for the final version of the book has been done by both authors.

January, 2000 Luis Ribes, Ottawa Pavel Zalesskii, Brasilia

Table of Contents

Preface ..................................................... VII

1 Inverse and Direct Limits 1.1 Inverse or Projective Limits ............................ 1 1.2 Direct or Inductive Limits ............................. 14 1.3 Notes, Comments and Further Reading .................. 18

2 Profinite Groups 2.1 Pro-C Groups ........................................ 19 2.2 Basic Properties of Pro - C Groups ...................... 28

Existence of Sections ............................ 29 Exactness of Inverse Limits of Profinite Groups ..... 31

2.3 The Order of a Profinite Group and Sylow Subgroups ..... 32 2.4 Generators ........................................... 43 2.5 Finitely Generated Profinite Groups .................... 45 2.6 Generators and Chains of Subgroups .................... 48 2.7 Procyclic Groups ..................................... 52 2.8 The Frattini Subgroup of a Profinite Group .............. 54 2.9 Pontryagin Duality for Profinite Groups ................. 60 2.10 Pullbacks and Pushouts ............................... 68 2.11 Profinite Groups as Galois Groups ...................... 70 2.12 Notes, Comments and Further Reading .................. 75

3 Free Profinite Groups 3.1 Profinite Topologies ................................... 79 3.2 The Pro-C Completion ................................ 82

The Completion Functor ........................ 85 3.3 Free Pro - C Groups ................................... 90

Free Pro - C Group on a Set Converging to 1 ....... 93 3.4 Maximal Pro - C Quotient Groups ....................... 101 3.5 Characterization of Free Pro-C Groups .................. 103 3.6 Open Subgroups of Free Pro - C Groups .................. 117 3.7 Notes, Comments and Further Reading .................. 120

XlI Table of Contents

4 Some Special Profinite Groups

4.1 Powers of Elements with Exponents from Z .............. 123 4.2 Subgroups of Finite Index in a Profinite Group ........... 124 4.3 Profinite Abelian Groups .............................. 133 4.4 Automorphism Group of a Profinite Group .............. 136 4.5 Automorphism Group of a Free Pro-p Group ............. 143 4.6 Profinite Frobenius Groups ............................ 147 4.7 Torsion in the Profinite Completion of a Group ........... 154 4.8 Notes, Comments and Further Reading .................. 160

5 Discrete and Profinite Modules 5.1 Profinite Rings and Modules ........................... 165

Duality Between Discrete and Profinite Modules .... 171 5.2 Free Profinite Modules ................................ 172 5.3 G-modules and Complete Group Algebras ............... 175

The Complete Group Algebra .................... 177 5.4 Projective and Injective Modules ....................... 179 5.5 Complete Tensor Products ............................. 183 5.6 Profinite G-spaces .................................... 187 5.7 Free Profinite [RGlmodules ........................... 195 5.8 Diagonal Actions ..................................... 196 5.9 Notes, Comments and Further Reading .................. 199

6 Homology and Cohomology of Profinite Groups 6.1 Review of Homological Algebra ......................... 201

Right and Left Derived Functors ................. 205 Bifunctors ..................................... 206 The Ext Functors ............................... 207 The Tor Functors ............................... 208

6.2 Cohomology with Coefficients in DMod([RGD) ........... 210 Standard Resolutions ........................... 211

6.3 Homology with Coefficients in PMod([RG]) ............. 214 6.4 Cohomology Groups with Coefficients in DMod(G) ....... 219 6.5 The Functorial Behavior of Hn(G,A) and Hn(G,A) ....... 220

The Inflation Map .............................. 221 6.6 Hn(G, A) as Derived Functors on DMod(G) ............. 227 6.7 Special Mappings ..................................... 231

The Restriction Map in Cohomology .............. 231 The Corestriction Map in Cohomology ............ 232 The Corestriction Map in Homology .............. 236 The Restriction Map in Homology ................ 236

6.8 Homology and Cohomology Groups in Low Dimensions .... 238 H2(G, A) and Extensions of Profinite Groups ...... 240

Table of Contents XIII

6.9 Extensions of Profinite Groups with Abelian Kernel ....... 245 6.10 Induced and Coinduced Modules ........................ 250 6.11 The Induced Module Ind~(B) for H Open ............... 255 6.12 Notes, Comments and Further Reading .................. 257

7 Cohomological Dimension 7.1 Basic Properties of Dimension .......................... 259 7.2 The Lyndon-Hochschild-Serre Spectral Sequence .......... 264 7.3 Cohomological Dimension of Subgroups .................. 269 7.4 Cohomological Dimension of Normal Subgroups and

Quotients ............................................ 274 7.5 Groups G with cdp(G) ::; 1 ............................. 276 7.6 Projective Profinite Groups ............................ 280 7.7 Free Pro-p Groups and Cohomological Dimension ......... 284 7.8 Generators and Relators for Pro-p Groups ............... 287 7.9 Cup Products ........................................ 291 7.10 Notes, Comments and Further Reading .................. 298

8 Normal Subgroups of Free Pro-C Groups 8.1 Normal Subgroup Generated by a Subset of a Basis ....... 302 8.2 The S-rank .......................................... 304 8.3 Accessible Subgroups .................................. 310 8.4 Accessible Subgroups H with wo(F / H) < rank(F) ........ 315 8.5 Homogeneous Pro - C Groups ........................... 322 8.6 Normal Subgroups .................................... 335 8.7 Proper Open Subgroups of Normal Subgroups ............ 344 8.8 The Congruence Kernel of SL2 (Z) ...................... 349 8.9 Sufficient Conditions for Freeness ....................... 350 8.10 Characteristic Subgroups of Free Pro-C Groups .......... 357 8.11 Notes, Comments and Further Reading .................. 360

9 Free Constructions of Profinite Groups 9.1 Free Pro-C Products .................................. 361 9.2 Amalgamated Free Pro - C Products ..................... 375 9.3 Cohomological Characterizations of Amalgamated

Products ............................................ 383 9.4 Pro - C HNN-extensions ................................ 390 9.5 Notes, Comments and Further Reading .................. 397

Open Questions 401

Appendix Al Spectral Sequences .................................... 403 A2 Positive Spectral Sequences ............................ 405

XIV Table of Contents

A3 Spectral Sequence of a Filtered Complex ................. 409 A4 Spectral Sequences of a Double Complex ................ 411 A5 Notes, Comments and Further Reading .................. 413

Bibliography ................................................ 415

Index of Symbols

Index of Authors

425

429

Index of Terms ............................................. 431