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 Chapter 5: THE BASIC CONSTRUCTION 63 5.1 The SpaceU63 5.2 The Case of a Pre-Coxeter System 66 5.3 Sectors inU68 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: LannA'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 sum; 123 7.1 The Nerve of a Coxeter System 123 7.2 Geometric Realizations 126 7.3 A Cell Structure on sum; 128 7.4 Examples 132 7.
5 Fixed Posets and Fixed Subspaces 133 Chapter 8: THE ALGEBRAIC TOPOLOGY OFUAND OF sum; 136 8.1 The Homology ofU137 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 ofW159 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 ofU166 9.2 What Is sum; Simply Connected at Infinity? 170 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 sum; a Manifold? 197 10.7 Reflection Groups on Homology Manifolds 197 10.8 Generalized Homology Spheres and Polytopes 201 10.9 Virtual PoincarA'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 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: sum; IS CAT(0): THEOREMS OF GROMOV AND ZMOUSSONG 230 12.1 A Piecewise Euclidean Cell Structure on sum; 231 12.2 The Right-Angled Case 233 12.3 The General Case 234 12.4 The Visual Boundary of sum; 237 12.
5 Background on Word Hyperbolic Groups 238 12.6 When Is sum; CAT(-1)? 241 12.7 Free Abelian Subgroups of Coxeter Groups 245 12.8 Relative Hyperbolization 247 Chapter 13: RIGIDITY 255 13.1 Definitions, Exa.