内容

目次

Preface xiii Chapter 1: INTRODUCTION AND PREVIEW 1 1.1Introduction 1 1.2 A Preview of the Right-Angled Case 9 Chapter 2: SOMEBASIC NOTIONS IN GEOMETRIC GROUP THEORY 15 2.1 Cayley Graphs and WordMetrics 15 2.2 Cayley 2-Complexes 18 2.3 Background on Aspherical Spaces 21Chapter 3: COXETER GROUPS 26 3.1 Dihedral Groups 26 3.2 ReflectionSystems 30 3.3 Coxeter Systems 37 3.4 The Word Problem 40 3.5 CoxeterDiagrams 42 Chapter 4: MORE COMBINATORIAL THEORY OF COXETER GROUPS 444.1 Special Subgroups in Coxeter Groups 44 4.2 Reflections 46 4.3 TheShortest Element in a Special Coset 47 4.4 Another Characterization ofCoxeter Groups 48 4.5 Convex Subsets of W 49 4.6 The Element of LongestLength 51 4.7 The Letters with Which a Reduced Expression Can End 53 4.8 ALemma of Tits 55 4.9 Subgroups Generated by Reflections 57 4.10 Normalizersof Special Subgroups 59 Chapter 5: THE BASIC CONSTRUCTION 63 5.1 TheSpace U 63 5.2 The Case of a Pre-Coxeter System 66 5.3 Sectors in U 68Chapter 6: GEOMETRIC REFLECTION GROUPS 72 6.1 Linear Reflections 73 6.2Spaces of Constant Curvature 73 6.3 Polytopes with Nonobtuse Dihedral Angles78 6.4 The Developing Map 81 6.5 Polygon Groups 85 6.6 Finite LinearGroups Generated by Reflections 87 6.7 Examples of Finite Reflection Groups92 6.8 Geometric Simplices: The Gram Matrix and the Cosine Matrix 96 6.9Simplicial Coxeter Groups: Lanni? er's Theorem 102 6.10 Three-dimensionalHyperbolic Reflection Groups: Andreev's Theorem 103 6.11 Higher-dimensionalHyperbolic Reflection Groups: Vinberg's Theorem 110 6.12 The CanonicalRepresentation 115 Chapter 7: THE COMPLEX ∑ 123 7.1 The Nerve of aCoxeter 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 133Chapter 8: THE ALGEBRAIC TOPOLOGY OF U AND OF ∑ 136 8.1 The Homology ofU 137 8.2 Acyclicity Conditions 140 8.3 Cohomology with Compact Supports146 8.4 The Case Where X Is a General Space 150 8.5 Cohomology with GroupRing Coefficients 152 8.6 Background on the Ends of a Group 157 8.7 TheEnds of W 159 8.8 Splittings of Coxeter Groups 160 8.9 Cohomology ofNormalizers of Spherical Special Subgroups 163 Chapter 9: THEFUNDAMENTAL GROUP AND THE FUNDAMENTAL GROUP AT INFINITY 166 9.1 TheFundamental Group of U 166 9.2 What Is ∑ Simply Connected at Infinity?170 Chapter 10: ACTIONS ON MANIFOLDS 176 10.1 Reflection Groups onManifolds 177 10.2 The Tangent Bundle 183 10.3 Background on ContractibleManifolds 185 10.4 Background on Homology Manifolds 191 10.5 AsphericalManifolds Not Covered by Euclidean Space 195 10.6 When Is ∑ a Manifold?197 10.7 Reflection Groups on Homology Manifolds 197 10.8 GeneralizedHomology Spheres and Polytopes 201 10.9 Virtual Poincari? e Duality Groups205 Chapter 11: THE REFLECTION GROUP TRICK 212 11.1 The First Versionof the Trick 212 11.2 Examples of Fundamental Groups of Closed AsphericalManifolds 215 11.3 Nonsmoothable Aspherical Manifolds 216 11.4 The BorelConjecture and the PDn-Group Conjecture 217 11.5 The Second Version of theTrick 220 11.6 The Bestvina-Brady Examples 222 11.7 The EquivariantReflection Group Trick 225 Chapter 12: ∑ IS CAT(0): THEOREMS OFGROMOV 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 TheVisual Boundary of ∑ 237 12.5 Background on Word Hyperbolic Groups 23812.6 When Is ∑ CAT(-1)? 241 12.7 Free Abelian Subgroups of CoxeterGroups 245 12.8 Relative Hyperbolization 247 Chapter 13: RIGIDITY 25513.1 Definitions, Examples, Counterexamples 255 13.2 Spherical ParabolicSubgroups and Their Fixed Subspaces 260 13.3 Coxeter Groups of Type PM 26313.4 Strong Rigidity for Groups of Type PM 268 Chapter 14: FREEQUOTIENTS AND SURFACE SUBGROUPS 276 14.1 Largeness 276 14.2 SurfaceSubgroups 282 Chapter 15: ANOTHER LOOK AT (CO)HOMOLOGY 286 15.1Cohomology with Constant Coefficients 286 15.2 Decompositions of CoefficientSystems 288 15.3 The W-Module Structure on (Co)homology 295 15.4 The CaseWhere W Is finite 303 Chapter 16: THE EULER CHARACTERISTIC 306 16.1Background on Euler Characteristics 306 16.2 The Euler CharacteristicConjecture 310 16.3 The Flag Complex Conjecture 313 Chapter 17: GROWTHSERIES 315 17.1 Rationality of the Growth Series 315 17.2 Exponentialversus Polynomial Growth 322 17.3 Reciprocity 324 17.4 Relationship withthe h-Polynomial 325 Chapter 18: BUILDINGS 328 18.1 The CombinatorialTheory of Buildings 328 18.2 The Geometric Realization of a Building 33618.3 Buildings Are CAT(0) 338 18.4 Euler-Poincari? e Measure 341Chapter 19: HECKE-VON NEUMANN ALGEBRAS 344 19.1 Hecke Algebras 344 19.2Hecke-Von Neumann Algebras 349 Chapter 20: WEIGHTED L2-(CO)HOMOLOGY 35920.1 Weighted L2-(Co)homology 361 20.2 Weighted L2-Betti Numbers and EulerCharacteristics 366 20.3 Concentration of (Co)homology in Dimension 0 36820.4 Weighted Poincari? e Duality 370 20.5 A Weighted Version of the SingerConjecture 374 20.6 Decomposition Theorems 376 20.7 Decoupling Cohomology389 20.8 L2-Cohomology of Buildings 394 Appendix A: CELL COMPLEXES 401A.1 Cells and Cell Complexes 401 A.2 Posets and Abstract SimplicialComplexes 406 A.3 Flag Complexes and Barycentric Subdivisions 409 A.4 Joins412 A.5 Faces and Cofaces 415 A.6 Links 418 Appendix B: REGULARPOLYTOPES 421 B.1 Chambers in the Barycentric Subdivision of a Polytope 421B.2 Classification of Regular Polytopes 424 B.3 Regular Tessellations ofSpheres 426 B.4 Regular Tessellations 428 Appendix C: THECLASSIFICATION OF SPHERICAL AND EUCLIDEAN COXETER GROUPS 433 C.1 Statementsof the Classification Theorems 433 C.2 Calculating Some Determinants 434C.3 Proofs of the Classification Theorems 436 Appendix D: THE GEOMETRICREPRESENTATION 439 D.1 Injectivity of the Geometric Representation 439 D.2The 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.2Complexes of Groups 454 E.3 The Meyer-Vietoris Spectral Sequence 459Appendix F: HOMOLOGY AND COHOMOLOGY OF GROUPS 465 F.1 Some Basic Definitions465 F.2 Equivalent (Co)homology with Group Ring Coefficients 467 F.3Cohomological Dimension and Geometric Dimension 470 F.4 FinitenessConditions 471 F.5 Poincari? e Duality Groups and Duality Groups 474Appendix G: ALGEBRAIC TOPOLOGY AT INFINITY 477 G.1 Some Algebra 477 G.2Homology and Cohomology at Infinity 479 G.3 Ends of a Space 482 G.4Semistability and the Fundamental Group at Infinity 483 Appendix H: THENOVIKOV AND BOREL CONJECTURES 487 H.1 Around the Borel Conjecture 487 H.2Smoothing Theory 491 H.3 The Surgery Exact Sequence and the Assembly MapConjecture 493 H.4 The Novikov Conjecture 496 Appendix I: NONPOSITIVECURVATURE 499 I.1 Geodesic Metric Spaces 499 I.2 The CAT(?)-Inequality 499I.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 516I.7 Moussong's Lemma 520 I.8 The Visual Boundary of a CAT(0)-Space 524Appendix J: L2-(CO)HOMOLOGY 531 J.1 Background on von Neumann Algebras 531J.2 The Regular Representation 531 J.3 L2-(Co)homology 538 J.4 Basic L2Algebraic Topology 541 J.5 L2-Betti Numbers and Euler Characteristics 544J.6 Poincari? e Duality 546 J.7 The Singer Conjecture 547 J.8 VanishingTheorems 548 Bibliography 555 Index 573