Cover -- Contents -- Preface -- Chapter 1 INTRODUCTION AND PREVIEW -- 1.1 Introduction -- 1.2 A Preview of the Right-Angled Case -- Chapter 2 SOME BASIC NOTIONS IN GEOMETRIC GROUP THEORY -- 2.1 Cayley Graphs and Word Metrics -- 2.2 Cayley 2-Complexes -- 2.3 Background on Aspherical Spaces -- Chapter 3 COXETER GROUPS -- 3.1 Dihedral Groups -- 3.2 Reflection Systems -- 3.3 Coxeter Systems -- 3.4 The Word Problem -- 3.5 Coxeter Diagrams -- Chapter 4 MORE COMBINATORIAL THEORY OF COXETER GROUPS -- 4.1 Special Subgroups in Coxeter Groups -- 4.2 Reflections -- 4.3 The Shortest Element in a Special Coset -- 4.4 Another Characterization of Coxeter Groups -- 4.5 Convex Subsets of W -- 4.6 The Element of Longest Length -- 4.7 The Letters with Which a Reduced Expression Can End -- 4.8 A Lemma of Tits -- 4.9 Subgroups Generated by Reflections -- 4.10 Normalizers of Special Subgroups -- Chapter 5 THE BASIC CONSTRUCTION -- 5.1 The Space U -- 5.2 The Case of a Pre-Coxeter System -- 5.3 Sectors in U -- Chapter 6 GEOMETRIC REFLECTION GROUPS -- 6.1 Linear Reflections -- 6.2 Spaces of Constant Curvature -- 6.3 Polytopes with Nonobtuse Dihedral Angles -- 6.4 The Developing Map -- 6.5 Polygon Groups -- 6.6 Finite Linear Groups Generated by Reflections -- 6.7 Examples of Finite Reflection Groups -- 6.8 Geometric Simplices: The Gram Matrix and the Cosine Matrix -- 6.9 Simplicial Coxeter Groups: Lannér's Theorem -- 6.10 Three-dimensional Hyperbolic Reflection Groups: Andreev's Theorem -- 6.11 Higher-dimensional Hyperbolic Reflection Groups: Vinberg's Theorem -- 6.12 The Canonical Representation -- Chapter 7 THE COMPLEX Σ -- 7.1 The Nerve of a Coxeter System -- 7.2 Geometric Realizations -- 7.3 A Cell Structure on Σ -- 7.4 Examples -- 7.5 Fixed Posets and Fixed Subspaces -- Chapter 8 THE ALGEBRAIC TOPOLOGY OF U AND OF Σ -- 8.1 The Homology of U.

8.2 Acyclicity Conditions -- 8.3 Cohomology with Compact Supports -- 8.4 The Case Where X Is a General Space -- 8.5 Cohomology with Group Ring Coefficients -- 8.6 Background on the Ends of a Group -- 8.7 The Ends of W -- 8.8 Splittings of Coxeter Groups -- 8.9 Cohomology of Normalizers of Spherical Special Subgroups -- Chapter 9 THE FUNDAMENTAL GROUP AND THE FUNDAMENTAL GROUP AT INFINITY -- 9.1 The Fundamental Group of U -- 9.2 What Is Σ Simply Connected at Infinity? -- Chapter 10 ACTIONS ON MANIFOLDS -- 10.1 Reflection Groups on Manifolds -- 10.2 The Tangent Bundle -- 10.3 Background on Contractible Manifolds -- 10.4 Background on Homology Manifolds -- 10.5 Aspherical Manifolds Not Covered by Euclidean Space -- 10.6 When Is Σ a Manifold? -- 10.7 Reflection Groups on Homology Manifolds -- 10.8 Generalized Homology Spheres and Polytopes -- 10.9 Virtual Poincaré Duality Groups -- Chapter 11 THE REFLECTION GROUP TRICK -- 11.1 The First Version of the Trick -- 11.2 Examples of Fundamental Groups of Closed Aspherical Manifolds -- 11.3 Nonsmoothable Aspherical Manifolds -- 11.4 The Borel Conjecture and the PD[sup(n)]-Group Conjecture -- 11.5 The Second Version of the Trick -- 11.6 The Bestvina-Brady Examples -- 11.7 The Equivariant Reflection Group Trick -- Chapter 12 Σ IS CAT(O): THEOREMS OF GROMOV AND MOUSSONG -- 12.1 A Piecewise Euclidean Cell Structure on Σ -- 12.2 The Right-Angled Case -- 12.3 The General Case -- 12.4 The Visual Boundary of Σ -- 12.5 Background on Word Hyperbolic Groups -- 12.6 When Is Σ CAT(-1)? -- 12.7 Free Abelian Subgroups of Coxeter Groups -- 12.8 Relative Hyperbolization -- Chapter 13 RIGIDITY -- 13.1 Definitions, Examples, Counterexamples -- 13.2 Spherical Parabolic Subgroups and Their Fixed Subspaces -- 13.3 Coxeter Groups of Type PM -- 13.4 Strong Rigidity for Groups of Type PM.

Chapter 14 FREE QUOTIENTS AND SURFACE SUBGROUPS -- 14.1 Largeness -- 14.2 Surface Subgroups -- Chapter 15 ANOTHER LOOK AT (CO)HOMOLOGY -- 15.1 Cohomology with Constant Coefficients -- 15.2 Decompositions of Coefficient Systems -- 15.3 The W-Module Structure on (Co)homology -- 15.4 The Case Where W Is finite -- Chapter 16 THE EULER CHARACTERISTIC -- 16.1 Background on Euler Characteristics -- 16.2 The Euler Characteristic Conjecture -- 16.3 The Flag Complex Conjecture -- Chapter 17 GROWTH SERIES -- 17.1 Rationality of the Growth Series -- 17.2 Exponential versus Polynomial Growth -- 17.3 Reciprocity -- 17.4 Relationship with the h-Polynomial -- Chapter 18 BUILDINGS -- 18.1 The Combinatorial Theory of Buildings -- 18.2 The Geometric Realization of a Building -- 18.3 Buildings Are CAT(0) -- 18.4 Euler-Poincaré Measure -- Chapter 19 HECKE-VON NEUMANN ALGEBRAS -- 19.1 Hecke Algebras -- 19.2 Hecke-Von Neumann Algebras -- Chapter 20 WEIGHTED L[sup(2)]-(CO)HOMOLOGY -- 20.1 Weighted L[sup(2)]-(Co)homology -- 20.2 Weighted L[sup(2)]-Betti Numbers and Euler Characteristics -- 20.3 Concentration of (Co)homology in Dimension 0 -- 20.4 Weighted Poincaré Duality -- 20.5 A Weighted Version of the Singer Conjecture -- 20.6 Decomposition Theorems -- 20.7 Decoupling Cohomology -- 20.8 L[sup(2)]-Cohomology of Buildings -- Appendix A: CELL COMPLEXES -- A.1 Cells and Cell Complexes -- A.2 Posets and Abstract Simplicial Complexes -- A.3 Flag Complexes and Barycentric Subdivisions -- A.4 Joins -- A.5 Faces and Cofaces -- A.6 Links -- Appendix B: REGULAR POLYTOPES -- B.1 Chambers in the Barycentric Subdivision of a Polytope -- B.2 Classification of Regular Polytopes -- B.3 Regular Tessellations of Spheres -- B.4 Regular Tessellations -- Appendix C: THE CLASSIFICATION OF SPHERICAL AND EUCLIDEAN COXETER GROUPS -- C.1 Statements of the Classification Theorems.

C.2 Calculating Some Determinants -- C.3 Proofs of the Classification Theorems -- Appendix D: THE GEOMETRIC REPRESENTATION -- D.1 Injectivity of the Geometric Representation -- D.2 The Tits Cone -- D.3 Complement on Root Systems -- Appendix E: COMPLEXES OF GROUPS -- E.1 Background on Graphs of Groups -- E.2 Complexes of Groups -- E.3 The Meyer-Vietoris Spectral Sequence -- Appendix F: HOMOLOGY AND COHOMOLOGY OF GROUPS -- F.1 Some Basic Definitions -- F.2 Equivalent (Co)homology with Group Ring Coefficients -- F.3 Cohomological Dimension and Geometric Dimension -- F.4 Finiteness Conditions -- F.5 Poincaré Duality Groups and Duality Groups -- Appendix G: ALGEBRAIC TOPOLOGY AT INFINITY -- G.1 Some Algebra -- G.2 Homology and Cohomology at Infinity -- G.3 Ends of a Space -- G.4 Semistability and the Fundamental Group at Infinity -- Appendix H: THE NOVIKOV AND BOREL CONJECTURES -- H.1 Around the Borel Conjecture -- H.2 Smoothing Theory -- H.3 The Surgery Exact Sequence and the Assembly Map Conjecture -- H.4 The Novikov Conjecture -- Appendix I: NONPOSITIVE CURVATURE -- I.1 Geodesic Metric Spaces -- I.2 The CAT(κ)-Inequality -- I.3 Polyhedra of Piecewise Constant Curvature -- I.4 Properties of CAT(0) Groups -- I.5 Piecewise Spherical Polyhedra -- I.6 Gromov's Lemma -- I.7 Moussong's Lemma -- I.8 The Visual Boundary of a CAT(0)-Space -- Appendix J: L[sup(2)]-(CO)HOMOLOGY -- J.1 Background on von Neumann Algebras -- J.2 The Regular Representation -- J.3 L[sup(2)]-(Co)homology -- J.4 Basic L[sup(2)] Algebraic Topology -- J.5 L[sup(2)]-Betti Numbers and Euler Characteristics -- J.6 Poincaré Duality -- J.7 The Singer Conjecture -- J.8 Vanishing Theorems -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- X -- Z.