The Geometry and Topology of Coxeter Groups. (LMS-32) available in Hardcover, eBook
The Geometry and Topology of Coxeter Groups. (LMS-32)
- ISBN-10:
- 0691131384
- ISBN-13:
- 9780691131382
- Pub. Date:
- 11/18/2007
- Publisher:
- Princeton University Press
- ISBN-10:
- 0691131384
- ISBN-13:
- 9780691131382
- Pub. Date:
- 11/18/2007
- Publisher:
- Princeton University Press
The Geometry and Topology of Coxeter Groups. (LMS-32)
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Overview
Product Details
ISBN-13: | 9780691131382 |
---|---|
Publisher: | Princeton University Press |
Publication date: | 11/18/2007 |
Series: | London Mathematical Society Monographs , #32 |
Edition description: | New Edition |
Pages: | 600 |
Product dimensions: | 6.00(w) x 9.25(h) x (d) |
Age Range: | 18 Years |
About the Author
Table of Contents
Preface xiii
Chapter 1: INTRODUCTION AND PREVIEW 11.1 Introduction 11.2 A Preview of the Right-Angled Case 9
Chapter 2: SOME BASIC NOTIONS IN GEOMETRIC GROUPTHEORY 152.1 Cayley Graphs and Word Metrics 152.2 Cayley 2-Complexes 182.3 Background on Aspherical Spaces 21
Chapter 3: COXETER GROUPS 263.1 Dihedral Groups 263.2 Reflection Systems 303.3 Coxeter Systems 373.4 The Word Problem 403.5 Coxeter Diagrams 42
Chapter 4: MORE COMBINATORIAL THEORY OF COXETERGROUPS 444.1 Special Subgroups in Coxeter Groups 444.2 Reflections 464.3 The Shortest Element in a Special Coset 474.4 Another Characterization of Coxeter Groups 484.5 Convex Subsets of W 494.6 The Element of Longest Length 514.7 The Letters with Which a Reduced Expression Can End 534.8 A Lemma of Tits 554.9 Subgroups Generated by Reflections 574.10 Normalizers of Special Subgroups 59
Chapter 5: THE BASIC CONSTRUCTION 635.1 The Space U 635.2 The Case of a Pre-Coxeter System 665.3 Sectors in U 68
Chapter 6: GEOMETRIC REFLECTION GROUPS 726.1 Linear Reflections 736.2 Spaces of Constant Curvature 736.3 Polytopes with Nonobtuse Dihedral Angles 786.4 The Developing Map 816.5 Polygon Groups 856.6 Finite Linear Groups Generated by Reflections 876.7 Examples of Finite Reflection Groups 926.8 Geometric Simplices: The Gram Matrix and the Cosine Matrix 966.9 Simplicial Coxeter Groups: Lanńer's Theorem 1026.10 Three-dimensional Hyperbolic Reflection Groups: Andreev's Theorem 1036.11 Higher-dimensional Hyperbolic Reflection Groups: Vinberg's Theorem 1106.12 The Canonical Representation 115
Chapter 7: THE COMPLEX ∑ 1237.1 The Nerve of a Coxeter System 1237.2 Geometric Realizations 1267.3 A Cell Structure on ∑ 1287.4 Examples 1327.5 Fixed Posets and Fixed Subspaces 133
Chapter 8: THE ALGEBRAIC TOPOLOGY OF U AND OF ∑ 1368.1 The Homology of U 1378.2 Acyclicity Conditions 1408.3 Cohomology with Compact Supports 1468.4 The Case Where X Is a General Space 1508.5 Cohomology with Group Ring Coefficients 1528.6 Background on the Ends of a Group 1578.7 The Ends of W 1598.8 Splittings of Coxeter Groups 1608.9 Cohomology of Normalizers of Spherical Special Subgroups 163
Chapter 9: THE FUNDAMENTAL GROUP AND THE FUNDAMENTALGROUP AT INFINITY 1669.1 The Fundamental Group of U 1669.2 What Is ∑ Simply Connected at Infinity? 170
Chapter 10: ACTIONS ON MANIFOLDS 17610.1 Reflection Groups on Manifolds 17710.2 The Tangent Bundle 18310.3 Background on Contractible Manifolds 18510.4 Background on Homology Manifolds 19110.5 Aspherical Manifolds Not Covered by Euclidean Space 19510.6 When Is ∑ a Manifold? 19710.7 Reflection Groups on Homology Manifolds 19710.8 Generalized Homology Spheres and Polytopes 20110.9 Virtual Poincaré Duality Groups 205
Chapter 11: THE REFLECTION GROUP TRICK 21211.1 The First Version of the Trick 21211.2 Examples of Fundamental Groups of Closed AsphericalManifolds 21511.3 Nonsmoothable Aspherical Manifolds 21611.4 The Borel Conjecture and the PDn-Group Conjecture 21711.5 The Second Version of the Trick 22011.6 The Bestvina-Brady Examples 22211.7 The Equivariant Reflection Group Trick 225
Chapter 12: ∑ IS CAT(0): THEOREMS OF GROMOV AND ZMOUSSONG 23012.1 A Piecewise Euclidean Cell Structure on ∑ 23112.2 The Right-Angled Case 23312.3 The General Case 23412.4 The Visual Boundary of ∑ 23712.5 Background on Word Hyperbolic Groups 23812.6 When Is ∑ CAT(-1)? 24112.7 Free Abelian Subgroups of Coxeter Groups 24512.8 Relative Hyperbolization 247
Chapter 13: RIGIDITY 25513.1 Definitions, Examples, Counterexamples 25513.2 Spherical Parabolic Subgroups and Their Fixed Subspaces 26013.3 Coxeter Groups of Type PM 26313.4 Strong Rigidity for Groups of Type PM 268
Chapter 14: FREE QUOTIENTS AND SURFACE SUBGROUPS 27614.1 Largeness 27614.2 Surface Subgroups 282
Chapter 15: ANOTHER LOOK AT (CO)HOMOLOGY 28615.1 Cohomology with Constant Coefficients 28615.2 Decompositions of Coefficient Systems 28815.3 The W-Module Structure on (Co)homology 29515.4 The Case Where W Is finite 303
Chapter 16: THE EULER CHARACTERISTIC 30616.1 Background on Euler Characteristics 30616.2 The Euler Characteristic Conjecture 31016.3 The Flag Complex Conjecture 313
Chapter 17: GROWTH SERIES 31517.1 Rationality of the Growth Series 31517.2 Exponential versus Polynomial Growth 32217.3 Reciprocity 32417.4 Relationship with the h-Polynomial 325
Chapter 18: BUILDINGS 32818.1 The Combinatorial Theory of Buildings 32818.2 The Geometric Realization of a Building 33618.3 Buildings Are CAT(0) 33818.4 Euler-Poincaré Measure 341
Chapter 19: HECKE-VON NEUMANN ALGEBRAS 34419.1 Hecke Algebras 34419.2 Hecke-Von Neumann Algebras 349
Chapter 20: WEIGHTED L2-(CO)HOMOLOGY 35920.1 Weighted L2-(Co)homology 36120.2 Weighted L2-Betti Numbers and Euler Characteristics 36620.3 Concentration of (Co)homology in Dimension 0 36820.4 Weighted Poincaré Duality 37020.5 A Weighted Version of the Singer Conjecture 37420.6 Decomposition Theorems 37620.7 Decoupling Cohomology 38920.8 L2-Cohomology of Buildings 394
Appendix A: CELL COMPLEXES 401A.1 Cells and Cell Complexes 401A.2 Posets and Abstract Simplicial Complexes 406A.3 Flag Complexes and Barycentric Subdivisions 409A.4 Joins 412A.5 Faces and Cofaces 415A.6 Links 418
Appendix B: REGULAR POLYTOPES 421B.1 Chambers in the Barycentric Subdivision of a Polytope 421B.2 Classification of Regular Polytopes 424B.3 Regular Tessellations of Spheres 426B.4 Regular Tessellations 428
Appendix C: THE CLASSIFICATION OF SPHERICAL AND EUCLIDEAN COXETER GROUPS 433C.1 Statements of the Classification Theorems 433C.2 Calculating Some Determinants 434C.3 Proofs of the Classification Theorems 436
Appendix D: THE GEOMETRIC REPRESENTATION 439D.1 Injectivity of the Geometric Representation 439D.2 The Tits Cone 442D.3 Complement on Root Systems 446
Appendix E: COMPLEXES OF GROUPS 449E.1 Background on Graphs of Groups 450E.2 Complexes of Groups 454E.3 The Meyer-Vietoris Spectral Sequence 459
Appendix F: HOMOLOGY AND COHOMOLOGY OF GROUPS 465F.1 Some Basic Definitions 465F.2 Equivalent (Co)homology with Group Ring Coefficients 467F.3 Cohomological Dimension and Geometric Dimension 470F.4 Finiteness Conditions 471F.5 Poincaré Duality Groups and Duality Groups 474
Appendix G: ALGEBRAIC TOPOLOGY AT INFINITY 477G.1 Some Algebra 477G.2 Homology and Cohomology at Infinity 479G.3 Ends of a Space 482G.4 Semistability and the Fundamental Group at Infinity 483
Appendix H: THE NOVIKOV AND BOREL CONJECTURES 487H.1 Around the Borel Conjecture 487H.2 Smoothing Theory 491H.3 The Surgery Exact Sequence and the Assembly Map Conjecture 493H.4 The Novikov Conjecture 496
Appendix I: NONPOSITIVE CURVATURE 499I.1 Geodesic Metric Spaces 499I.2 The CAT(?)-Inequality 499I.3 Polyhedra of Piecewise Constant Curvature 507I.4 Properties of CAT(0) Groups 511I.5 Piecewise Spherical Polyhedra 513I.6 Gromov's Lemma 516I.7 Moussong's Lemma 520I.8 The Visual Boundary of a CAT(0)-Space 524
Appendix J: L2-(CO)HOMOLOGY 531J.1 Background on von Neumann Algebras 531J.2 The Regular Representation 531J.3 L2-(Co)homology 538J.4 Basic L2 Algebraic Topology 541J.5 L2-Betti Numbers and Euler Characteristics 544J.6 Poincaré Duality 546J.7 The Singer Conjecture 547J.8 Vanishing Theorems 548
Bibliography 555Index 573
What People are Saying About This
This is a comprehensivenearly encyclopedicsurvey of results concerning Coxeter groups. No other book covers the more recent important results, many of which are due to Michael Davis himself. This is an excellent, thoughtful, and well-written book, and it should have a wide readership among pure mathematicians in geometry, topology, representation theory, and group theory.
Graham A. Niblo, University of Southampton
"This is a comprehensive—nearly encyclopedic—survey of results concerning Coxeter groups. No other book covers the more recent important results, many of which are due to Michael Davis himself. This is an excellent, thoughtful, and well-written book, and it should have a wide readership among pure mathematicians in geometry, topology, representation theory, and group theory."—Graham A. Niblo, University of Southampton"Davis's book is a significant addition to the mathematics literature and it provides an important access point for geometric group theory. Although the book is a focused research monograph, it does such a nice job of presenting important material that it will also serve as a reference for quite some time. In fact, for years to come mathematicians will be writing 'terminology and notation follow Davis' in the introductions to papers on the geometry and topology of infinite Coxeter groups."—John Meier, Lafayette College
Davis's book is a significant addition to the mathematics literature and it provides an important access point for geometric group theory. Although the book is a focused research monograph, it does such a nice job of presenting important material that it will also serve as a reference for quite some time. In fact, for years to come mathematicians will be writing 'terminology and notation follow Davis' in the introductions to papers on the geometry and topology of infinite Coxeter groups.
John Meier, Lafayette College