August 17, 2007 Time: 09:52am prelims.tex THE GEOMETRY AND TOPOLOGY OF COXETER GROUPS i August 17, 2007 Time: 09:52am prelims.tex 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. EDITORS: Martin Bridson, Imperial College, London, Terry Lyons, University of Oxford, and Peter Sarnak, Princeton University and Courant Institute, New York EDITORIAL ADVISERS: J. H. Coates, University of Cambridge, W. S. Kendall, University of Warwick, and J ´ anos Koll ´ ar, 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 ii August 17, 2007 Time: 09:52am prelims.tex THE GEOMETRY AND TOPOLOGY OF COXETER GROUPS Michael W. Davis PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD iii August 17, 2007 Time: 09:52am prelims.tex 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 L A T E X Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10987654321 iv August 17, 2007 Time: 09:52am prelims.tex To Wanda v August 17, 2007 Time: 09:52am prelims.tex vi August 17, 2007 Time: 09:52am prelims.tex 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 August 17, 2007 Time: 09:52am prelims.tex 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: Lann ´ er’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 August 17, 2007 Time: 09:52am prelims.tex 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 Poincar ´ e 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 PD n -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 August 17, 2007 Time: 09:52am prelims.tex 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-Poincar ´ e Measure 341 Chapter 19 HECKE–VON NEUMANN ALGEBRAS 344 19.1 Hecke Algebras 344 19.2 Hecke–Von Neumann Algebras 349 Chapter 20 WEIGHTED L 2 -(CO)HOMOLOGY 359 20.1 Weighted L 2 -(Co)homology 361 20.2 Weighted L 2 -Betti Numbers and Euler Characteristics 366 20.3 Concentration of (Co)homology in Dimension 0 368 20.4 Weighted Poincar ´ e Duality 370 20.5 A Weighted Version of the Singer C onjecture 374 20.6 Decomposition Theorems 376 20.7 Decoupling Cohomology 389 20.8 L 2 -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 [...]... reflection groups is important in Lie theory [29], in the theory of lattices in Rn and in E Cartan’s theory of symmetric spaces The classification of these groups and of the finite (spherical) reflection groups can be found in Coxeter s 1934 paper [67] We give this classification in Table 6.1 of Section 6.9 and its proof in Appendix C There are also examples of discrete groups generated by reflections on the other... be extracted from the first part of Bourbaki [29] It is also proved as the main result (Theorem 3.3.4) of Chapter 3 The equivalence of these two definitions is the principal mechanism driving the combinatorial theory of Coxeter groups The details of the second definition go as follows For each pair (s, t) ∈ S × S, let mst denote the order of st The matrix (mst ) is the Coxeter matrix of (W, S); it is a... 1-skeleton of an n-cube • If n = 3 and L is the disjoint union of a 1-simplex and a 0-simplex, then PL is the subcomplex of the 3-cube consisting of the top and bottom faces and the 4 vertical edges (See Figure 1.1.) n−1 , then PL = [−1, 1]n August 2, 2007 Time: 12:25pm chapter1.tex 11 INTRODUCTION AND PREVIEW L P L Figure 1.1 L is the union of a 1-simplex and a 0-simplex • Suppose L is the join of two... Finite groups generated by orthogonal linear reflections on Rn play a decisive role in • the classification of Lie groups and Lie algebras; • the theory of algebraic groups, as well as, the theories of spherical buildings and finite groups of Lie type; • the classification of regular polytopes (see [69, 74, 201] or Appendix B) Finite reflection groups also play important roles in many other areas of mathematics,... {a, b} and R = {aba−1 b−1 }, then G = C∞ × C∞ , the product of two infinite cyclic groups G can be identified with the integer lattice in R2 and Cay(G, S) with the grid consisting of the union of all horizontal and vertical lines through points with integral coordinates The complex is the cellulation of R2 obtained by filling in the squares In this case, is the same as X (the universal cover of the presentation... Classification Theorems Calculating Some Determinants Proofs of the Classification Theorems Appendix D THE GEOMETRIC REPRESENTATION D.1 D.2 D.3 Injectivity of the Geometric Representation The Tits Cone Complement on Root Systems Appendix E COMPLEXES OF GROUPS E.1 E.2 E.3 Background on Graphs of Groups Complexes of Groups The Meyer-Vietoris Spectral Sequence Appendix F HOMOLOGY AND COHOMOLOGY OF GROUPS F.1... a Coxeter system with nerve L So, the topological type of L is completely arbitrary This arbitrariness is the source of power for the using Coxeter groups to construct interesting examples in geometric and combinatorial group theory Coxeter Groups as a Source of Examples in Geometric and Combinatorial Group Theory Here are some of the examples • The Eilenberg-Ganea Problem asks if every group π of. .. which remedies these difficulties This alternative is the cell complex , discusssed below and in greater detail in Chapters 7 and 12 (and many other places throughout the book) The Cell Complex Given a Coxeter system (W, S), in Chapter 7 we construct a cell complex with the following properties: • The 0-skeleton of is W • The 1-skeleton of is Cay(W, S), the Cayley graph of 2.1 • The 2-skeleton of is a Cayley... to Coxeter groups do not appear in this book, such as the Bruhat order, root systems, Kazhdan–Lusztig polynomials, and the relationship of Coxeter groups to Lie theory The principal reason for their omission is my ignorance about them 1.2 A PREVIEW OF THE RIGHT-ANGLED CASE In the right-angled case the construction of simplifies considerably We describe it here In fact, this case is sufficient for the. .. singleton, say, S = {a} and R = {am }, for some m > 2 Then G is cyclic of order m, X is the result of gluing a 2-disk onto a circle via a degree-m map ∂D2 → S1 and X consists of m-copies of a 2-disk with their boundaries identified Their common boundary is the single circuit corresponding to the relation r = am On the other hand, the 2-complex is the single 2-disk Dr (an m-gon) with the cyclic group acting . prelims.tex THE GEOMETRY AND TOPOLOGY OF COXETER GROUPS Michael W. Davis PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD iii August 17, 2007 Time: 09:52am prelims.tex Copyright c  2008 by Princeton University. 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. 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