The Geometry and Topology of Coxeter Groups. (LMS-32)

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 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
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 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 ZMOUSSONG 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
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 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 Poincar¿e 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
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 AND COHOMOLOGY OF GROUPS 465
F.1 Some Basic Definitions 465
F.2 Equivalent (Co)homology with Group Ring Coefficients 467
F.3 Cohomological Dimension and Geometric Dimension 470
F.4 Finiteness Conditions 471
F.5 Poincar¿e Duality Groups and Duality Groups 474
Appendix G: ALGEBRAIC TOPOLOGY AT INFINITY 477
G.1 Some Algebra 477
G.2 Homology and Cohomology at Infinity 479
G.3 Ends of a Space 482
G.4 Semistability and the Fundamental Group at Infinity 483
Appendix H: THE NOVIKOV AND BOREL CONJECTURES 487
H.1 Around the Borel Conjecture 487
H.2 Smoothing Theory 491
H.3 The Surgery Exact Sequence and the Assembly Map Conjecture 493
H.4 The Novikov Conjecture 496
Appendix I: NONPOSITIVE CURVATURE 499
I.1 Geodesic Metric Spaces 499
I.2 The CAT(?)-Inequality 499
I.3 Polyhedra of Piecewise Constant Curvature 507
I.4 Properties of CAT(0) Groups 511
I.5 Piecewise Spherical Polyhedra 513
I.6 Gromov's Lemma 516
I.7 Moussong's Lemma 520
I.8 The Visual Boundary of a CAT(0)-Space 524
Appendix J: L²-(CO)HOMOLOGY 531
J.1 Background on von Neumann Algebras 531
J.2 The Regular Representation 531
J.3 L²-(Co)homology 538
J.4 Basic L² Algebraic Topology 541
J.5 L²-Betti Numbers and Euler Characteristics 544
J.6 Poincar¿e Duality 546
J.7 The Singer Conjecture 547
J.8 Vanishing Theorems 548
Bibliography 555
Index 573