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August 17, 2007 Time: 09:52am prelims.tex

THE GEOMETRY AND TOPOLOGY

OF COXETER GROUPS

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London Mathematical Society Monographs Series

The London Mathematical Society Monographs Series was established in 1968. Since that time it has published outstanding volumes that have been critically acclaimed by the mathematics community. The aim of this series is to publish authoritative accounts of current research in mathematics and high- quality expository works bringing the reader to the frontiers of research. Of particular interest are topics that have developed rapidly in the last ten years but that have reached a certain level of maturity. Clarity of exposition is important and each book should be accessible to those commencing work in its field.

The original series was founded in 1968 by the Society and Academic Press; the second series was launched by the Society and Oxford University Press in 1983. In January 2003, the Society and Princeton University Press united to expand the number of books published annually and to make the series more international in scope.

E D I T O R S : Martin Bridson, Imperial College, London, Terry Lyons, University of Oxford, and Peter Sarnak, Princeton University and Courant Institute, New York

E D I T O R I A L A D V I S E R S : J. H. Coates, University of Cambridge, W. S. Kendall, University of Warwick, and János Kollár, Princeton University

Vol. 32, The Geometry and Topology of Coxeter Groups by Michael W. Davis Vol. 31, Analysis of Heat Equations on Domains by El Maati Ouhabaz

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THE GEOMETRY AND TOPOLOGY OF COXETER GROUPS

Michael W. Davis

P R I N C E T O N U N I V E R S I T Y P R E S S

P R I N C E T O N A N D O X F O R D

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Copyright c© 2008 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY

All Rights Reserved

Library of Congress Cataloging-in-Publication Data Davis, Michael The geometry and topology of Coxeter groups / Michael W. Davis. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-691-13138-2 (alk. paper) ISBN-10: 0-691-13138-4 1. Coxeter groups. 2. Geometric group theory. I. Title. QA183.D38 2007 51s′.2–dc22 2006052879

British Library Cataloging-in-Publication Data is available This book has been composed in LATEX Printed on acid-free paper. ∞

press.princeton.edu

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

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To Wanda

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Contents

Preface xiii

Chapter 1 INTRODUCTION AND PREVIEW 1

1.1 Introduction 1 1.2 A Preview of the Right-Angled Case 9

Chapter 2 SOME BASIC NOTIONS IN GEOMETRIC GROUP THEORY 15

2.1 Cayley Graphs and Word Metrics 15 2.2 Cayley 2-Complexes 18 2.3 Background on Aspherical Spaces 21

Chapter 3 COXETER GROUPS 26

3.1 Dihedral Groups 26 3.2 Reflection Systems 30 3.3 Coxeter Systems 37 3.4 The Word Problem 40 3.5 Coxeter Diagrams 42

Chapter 4 MORE COMBINATORIAL THEORY OF COXETER GROUPS 44

4.1 Special Subgroups in Coxeter Groups 44 4.2 Reflections 46 4.3 The Shortest Element in a Special Coset 47 4.4 Another Characterization of Coxeter Groups 48 4.5 Convex Subsets of W 49 4.6 The Element of Longest Length 51 4.7 The Letters with Which a Reduced Expression Can End 53 4.8 A Lemma of Tits 55 4.9 Subgroups Generated by Reflections 57 4.10 Normalizers of Special Subgroups 59

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viii CONTENTS

Chapter 5 THE BASIC CONSTRUCTION 63

5.1 The Space U 63 5.2 The Case of a Pre-Coxeter System 66 5.3 Sectors in U 68

Chapter 6 GEOMETRIC REFLECTION GROUPS 72

6.1 Linear Reflections 73 6.2 Spaces of Constant Curvature 73 6.3 Polytopes with Nonobtuse Dihedral Angles 78 6.4 The Developing Map 81 6.5 Polygon Groups 85 6.6 Finite Linear Groups Generated by Reflections 87 6.7 Examples of Finite Reflection Groups 92 6.8 Geometric Simplices: The Gram Matrix and the Cosine Matrix 96 6.9 Simplicial Coxeter Groups: Lannér’s Theorem 102 6.10 Three-dimensional Hyperbolic Reflection Groups: Andreev’s

Theorem 103 6.11 Higher-dimensional Hyperbolic Reflection Groups: Vinberg’s

Theorem 110 6.12 The Canonical Representation 115

Chapter 7 THE COMPLEX � 123

7.1 The Nerve of a Coxeter System 123 7.2 Geometric Realizations 126 7.3 A Cell Structure on � 128 7.4 Examples 132 7.5 Fixed Posets and Fixed Subspaces 133

Chapter 8 THE ALGEBRAIC TOPOLOGY OF U AND OF � 136 8.1 The Homology of U 137 8.2 Acyclicity Conditions 140 8.3 Cohomology with Compact Supports 146 8.4 The Case Where X Is a General Space 150 8.5 Cohomology with Group Ring Coefficients 152 8.6 Background on the Ends of a Group 157 8.7 The Ends of W 159 8.8 Splittings of Coxeter Groups 160 8.9 Cohomology of Normalizers of Spherical Special Subgroups 163

Chapter 9 THE FUNDAMENTAL GROUP AND THE FUNDAMENTAL GROUP AT INFINITY 166

9.1 The Fundamental Group of U 166 9.2 What Is � Simply Connected at Infinity? 170

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CONTENTS ix

Chapter 10 ACTIONS ON MANIFOLDS 176

10.1 Reflection Groups on Manifolds 177 10.2 The Tangent Bundle 183 10.3 Background on Contractible Manifolds 185 10.4 Background on Homology Manifolds 191 10.5 Aspherical Manifolds Not Covered by Euclidean Space 195 10.6 When Is � a Manifold? 197 10.7 Reflection Groups on Homology Manifolds 197 10.8 Generalized Homology Spheres and Polytopes 201 10.9 Virtual Poincaré Duality Groups 205

Chapter 11 THE REFLECTION GROUP TRICK 212

11.1 The First Version of the Trick 212 11.2 Examples of Fundamental Groups of Closed Aspherical

Manifolds 215 11.3 Nonsmoothable Aspherical Manifolds 216 11.4 The Borel Conjecture and the PDn-Group Conjecture 217 11.5 The Second Version of the Trick 220 11.6 The Bestvina-Brady Examples 222 11.7 The Equivariant Reflection Group Trick 225

Chapter 12 � IS CAT(0): THEOREMS OF GROMOV AND MOUSSONG 230

12.1 A Piecewise Euclidean Cell Structure on � 231 12.2 The Right-Angled Case 233 12.3 The General Case 234 12.4 The Visual Boundary of � 237 12.5 Background on Word Hyperbolic Groups 238 12.6 When Is � CAT(−1)? 241 12.7 Free Abelian Subgroups of Coxeter Groups 245 12.8 Relative Hyperbolization 247

Chapter 13 RIGIDITY 255

13.1 Definitions, Examples, Counterexamples 255 13.2 Spherical Parabolic Subgroups and Their Fixed Subspaces 260 13.3 Coxeter Groups of Type PM 263 13.4 Strong Rigidity for Groups of Type PM 268

Chapter 14 FREE QUOTIENTS AND SURFACE SUBGROUPS 276

14.1 Largeness 276 14.2 Surface Subgroups 282

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x CONTENTS

Chapter 15 ANOTHER LOOK AT (CO)HOMOLOGY 286

15.1 Cohomology with Constant Coefficients 286 15.2 Decompositions of Coefficient Systems 288 15.3 The W-Module Structure on (Co)homology 295 15.4 The Case Where W Is finite 303

Chapter 16 THE EULER CHARACTERISTIC 306

16.1 Background on Euler Characteristics 306 16.2 The Euler Characteristic Conjecture 310 16.3 The Flag Complex Conjecture 313

Chapter 17 GROWTH SERIES 315

17.1 Rationality of the Growth Series 315 17.2 Exponential versus Polynomial Growth 322 17.3 Reciprocity 324 17.4 Relationship with the h-Polynomial 325

Chapter 18 BUILDINGS 328

18.1 The Combinatorial Theory of Buildings 328 18.2 The Geometric Realization of a Building 336 18.3 Buildings Are CAT(0) 338 18.4 Euler-Poincaré Measure 341

Chapter 19 HECKE–VON NEUMANN ALGEBRAS 344

19.1 Hecke Algebras 344 19.2 Hecke–Von Neumann Algebras 349

Chapter 20 WEIGHTED L2-(CO)HOMOLOGY 359

20.1 Weighted L2-(Co)homology 361 20.2 Weighted L2-Betti Numbers and Euler Characteristics 366 20.3 Concentration of (Co)homology in Dimension 0 368 20.4 Weighted Poincaré Duality 370 20.5 A Weighted Version of the Singer Conjecture 374 20.6 Decomposition Theorems 376 20.7 Decoupling Cohomology 389 20.8 L2-Cohomology of Buildings 394

Appendix A CELL COMPLEXES 401

A.1 Cells and Cell Complexes 401 A.2 Posets and Abstract Simplicial Complexes 406 A.3 Flag Complexes and Barycentric Subdivisions 409 A.4 Joins 412

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CONTENTS xi

A.5 Faces and Cofaces 415 A.6 Links 418

Appendix B REGULAR POLYTOPES 421

B.1 Chambers in the Barycentric Subdivision of a Polytope 421 B.2 Classification of Regular Polytopes 424 B.3 Regular Tessellations of Spheres 426 B.4 Regular Tessellations 428

Appendix C THE CLASSIFICATION OF SPHERICAL AND EUCLIDEAN COXETER GROUPS 433

C.1 Statements of the Classification Theorems 433 C.2 Calculating Some Determinants 434 C.3 Proofs of the Classification Theorems 436

Appendix D THE GEOMETRIC REPRESENTATION 439

D.1 Injectivity of the Geometric Representation 439 D.2 The Tits Cone 442 D.3 Complement on Root Systems 446

Appendix E COMPLEXES OF GROUPS 449

E.1 Background on Graphs of Groups 450 E.2 Complexes of Groups 454 E.3 The Meyer-Vietoris Spectral Sequence 459

Appendix F HOMOLOGY A