Verfügbarkeit: The geometry and topology of Coxeter groups

The geometry and topology of Coxeter groups:

Gespeichert in:

Bibliographische Detailangaben
1. Verfasser:	Davis, Michael (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Princeton, N. J. [u.a.] Princeton Univ. Press 2008
Schriftenreihe:	London Mathematical Society monographs series 32
Schlagworte:	Coxeter groups Geometric group theory Algebraische Topologie Geometrie Coxeter-Gruppe
Online-Zugang:	Publisher description Inhaltsverzeichnis
Beschreibung:	Includes bibliographical references and index
Beschreibung:	XIV, 584 S. graph. Darst.
ISBN:	0691131384 9780691131382

Internformat

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adam_text	Contents Preface xiü Chapter 1 INTRODUCTION AND PREVIEW 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 1 5 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. 1 0 Normalizers of Special Subgroups 59 VIH CONTENTS Chapter 5 THE BASIC CONSTRUCTION 63 5.Ì The Space If 63 5.2 The Case of a Pre-Coxeter System 66 5.3 Sectors in U 68 Chapter б 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 SimpHces: The Gram Matrix and the Cosine Matrix 96 6.9 Simplicia! Coxeter Groups: banner 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 Σ 123 7.1 The Nerve of a Coxeter System 12 3 7.2 Geometric Realizations 126 7.3 A Cell Structure on Σ 128 7.4 Examples 132 7.5 Fixed Posets and Fixed Subspaces 1 33 Chapter 8 THE ALGEBRAIC TOPOLOGY OF U AND OF Σ 136 8.1 TheHomoJogyofW 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 1 57 8.7 The Ends of W 159 8.8 Splittings of Coxeter Groups 160 8.9 Cohomology of Normalizers of Spherical Special Subgroups 1 63 Chapter 9 THE FUNDAMENTAL GROUP AND THE FUNDAMENTAL GROUP AT INFINITY 166 9. 1 The Fundamental Group of U 166 9.2 What Is Σ Simply Connected at Infinity? 1 70 CONTENTS ¡χ Chapter 1 0 ACTIONS ON MANIFOLDS 176 10.1 Reflection Groups on Manifolds 177 J 0.2 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 Σ a Manifold? 197 10.7 Reflection Groups on Homology Manifolds 197 10.8 Generalized Homology Spheres and Polytopes 201 10.9 Virtual Poincaré Duality Groups 205 Chapter 11 THE REFLECTION GROUP TRICK 212 1 ]. 1 The First Version of the Trick 212 11.2 Examples of Fundamental Groups of Closed Aspherical Manifolds 215 11.3 Nonsmoothable Aspherical Manifolds 21 б 11.4 The Borei Conjecture and the PD -Group Conjecture 21 7 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 1 2 Σ IS CAliO): THEOREMS OF GROMOV AND MOUSSONG 230 12.1 A Piecewise Euclidean Cell Structure on Σ 231 12.2 The Right-Angled Case 233 12.3 The General Case 234 12.4 The Visual Boundary of Σ 237 12.5 Background on Word Hyperbolic Groups 238 12.6 When Is Σ CATC- 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 X CONTENTS Chapter 1 5 ANOTHER LOOK AT (CO)HOMOLOGY 286 1 5.1 Cohomology with Constant Coefficients 286 15.2 Decompositions of Coefficient Systems 288 15.3 The W-Module Structure on (Cojhomology 295 15.4 The Case Where W Is finite 303 Chapter 1 6 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 1 7.1 Rationality of the Growth Series 31 5 17.2 Exponential versus Polynomial Growth 322 17.3 Reciprocity 324 17.4 Relationship with the /г -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(O) 338 18.4 Euler-Poincaré Measure 341 Chapter 19 HECKE-VON NEUMANN ALGEBRAS 344 19.1 Hecke Algebras 344 19.2 Hecke-Von Neumann Algebras 349 Chapter 20 WEIGHTED ^-(COJHOMOLOGY 359 20.1 Weighted ¿2-(Co)homology 361 20.2 Weighted /ЛВеМ Numbers and Euler Characteristics 366 20.3 Concentration of (Co)homology in Dimension 0 368 20.4 Weighted Poincaré Duality 370 20.5 A Weighted Version of the Singer Conjecture 374 20.6 Decomposition Theorems 376 20.7 Decoupling Cohomology 389 20.8 ^-Cohomology of Buildings 394 Appendix A CELL COMPLEXES 401 A.I 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 CONTENTS χ/ Α. 5 Faces and Cofaces 415 A. 6 Links 418 Appendix В REGULAR POLYTOPES 421 B.I 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 С THE CLASSIFICATION OF SPHERICAL AND EUCLIDEAN COXETER GROUPS 433 C.I 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 Poincaré Duality Groups and Duality Groups 474 Appendix G ALGEBRAIC TOPOLOGY AT INFINITY 477 G.I 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.I Around the Borei 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 XÜ CONTENTS Appendix I NONPOSITIVE CURVATURE 499 1. 1 Geodesic Metric Spaces 499 1.2 The CAT(«-)-lnequality 499 1.3 Polyhedra of Piecewise Constant Curvature 507 1.4 Properties of САДО) Groups 511 1.5 Piecewise Spherical Polyhedra 513 1.6 Gromov s Lemma 516 1.7 Moussong s Lemma 520 1.8 The Visual Boundary of a CAT(O)-Space 524 Appendix J ^-(COjHOMOLOGY 531 J.I 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 Poincaré Duality 546 J.7 The Singer Conjecture 547 J.8 Vanishing Theorems 548 Bibliography 555 Index 573
adam_txt	Contents Preface xiü Chapter 1 INTRODUCTION AND PREVIEW 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 1 5 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. 1 0 Normalizers of Special Subgroups 59 VIH CONTENTS Chapter 5 THE BASIC CONSTRUCTION 63 5.Ì The Space If 63 5.2 The Case of a Pre-Coxeter System 66 5.3 Sectors in U 68 Chapter б 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 SimpHces: The Gram Matrix and the Cosine Matrix 96 6.9 Simplicia! Coxeter Groups: banner'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 Σ 123 7.1 The Nerve of a Coxeter System 12 3 7.2 Geometric Realizations 126 7.3 A Cell Structure on Σ 128 7.4 Examples 132 7.5 Fixed Posets and Fixed Subspaces 1 33 Chapter 8 THE ALGEBRAIC TOPOLOGY OF U AND OF Σ 136 8.1 TheHomoJogyofW 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 1 57 8.7 The Ends of W 159 8.8 Splittings of Coxeter Groups 160 8.9 Cohomology of Normalizers of Spherical Special Subgroups 1 63 Chapter 9 THE FUNDAMENTAL GROUP AND THE FUNDAMENTAL GROUP AT INFINITY 166 9. 1 The Fundamental Group of U 166 9.2' What Is Σ Simply Connected at Infinity? 1 70 CONTENTS ¡χ Chapter 1 0 ACTIONS ON MANIFOLDS 176 10.1 Reflection Groups on Manifolds 177 J 0.2 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 Σ a Manifold? 197 10.7 Reflection Groups on Homology Manifolds 197 10.8 Generalized Homology Spheres and Polytopes 201 10.9 Virtual Poincaré Duality Groups 205 Chapter 11 THE REFLECTION GROUP TRICK 212 1 ]. 1 The First Version of the Trick 212 11.2 Examples of Fundamental Groups of Closed Aspherical Manifolds 215 11.3 Nonsmoothable Aspherical Manifolds 21 б 11.4 The Borei Conjecture and the PD"-Group Conjecture 21 7 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 1 2 Σ IS CAliO): THEOREMS OF GROMOV AND MOUSSONG 230 12.1 A Piecewise Euclidean Cell Structure on Σ 231 12.2 The Right-Angled Case 233 12.3 The General Case 234 12.4 The Visual Boundary of Σ 237 12.5 Background on Word Hyperbolic Groups 238 12.6 When Is Σ CATC- 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 X CONTENTS Chapter 1 5 ANOTHER LOOK AT (CO)HOMOLOGY 286 1 5.1 Cohomology with Constant Coefficients 286 15.2 Decompositions of Coefficient Systems 288 15.3 The W-Module Structure on (Cojhomology 295 15.4 The Case Where W Is finite 303 Chapter 1 6 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 1 7.1 Rationality of the Growth Series 31 5 17.2' Exponential versus Polynomial Growth 322 17.3 Reciprocity 324 17.4 Relationship with the /г -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(O) 338 18.4 Euler-Poincaré Measure 341 Chapter 19 HECKE-VON NEUMANN ALGEBRAS 344 19.1 Hecke Algebras 344 19.2 Hecke-Von Neumann Algebras 349 Chapter 20 WEIGHTED ^-(COJHOMOLOGY 359 20.1 Weighted ¿2-(Co)homology 361 20.2 Weighted /ЛВеМ Numbers and Euler Characteristics 366 20.3 Concentration of (Co)homology in Dimension 0 368 20.4 Weighted Poincaré Duality 370 20.5 A Weighted Version of the Singer Conjecture 374 20.6 Decomposition Theorems 376 20.7 Decoupling Cohomology 389 20.8' ^-Cohomology of Buildings 394 Appendix A CELL COMPLEXES 401 A.I 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 CONTENTS χ/ Α. 5 Faces and Cofaces 415 A. 6 Links 418 Appendix В REGULAR POLYTOPES 421 B.I 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 С THE CLASSIFICATION OF SPHERICAL AND EUCLIDEAN COXETER GROUPS 433 C.I 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 Poincaré Duality Groups and Duality Groups 474 Appendix G ALGEBRAIC TOPOLOGY AT INFINITY 477 G.I 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.I Around the Borei 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 XÜ CONTENTS Appendix I NONPOSITIVE CURVATURE 499 1. 1 Geodesic Metric Spaces 499 1.2 The CAT(«-)-lnequality 499 1.3 Polyhedra of Piecewise Constant Curvature 507 1.4 Properties of САДО) Groups 511 1.5 Piecewise Spherical Polyhedra 513 1.6 Gromov's Lemma 516 1.7 Moussong's Lemma 520 1.8 The Visual Boundary of a CAT(O)-Space 524 Appendix J ^-(COjHOMOLOGY 531 J.I 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 Poincaré Duality 546 J.7 The Singer Conjecture 547 J.8 Vanishing Theorems 548 Bibliography 555 Index 573
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publishDate	2008
publishDateSearch	2008
publishDateSort	2008
publisher	Princeton Univ. Press
record_format	marc
series	London Mathematical Society monographs series
series2	London Mathematical Society monographs series
spelling	Davis, Michael Verfasser aut The geometry and topology of Coxeter groups Michael W. Davis Princeton, N. J. [u.a.] Princeton Univ. Press 2008 XIV, 584 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society monographs series 32 Includes bibliographical references and index Coxeter groups Geometric group theory Algebraische Topologie (DE-588)4120861-4 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Coxeter-Gruppe (DE-588)4261522-7 gnd rswk-swf Coxeter-Gruppe (DE-588)4261522-7 s Geometrie (DE-588)4020236-7 s DE-604 Algebraische Topologie (DE-588)4120861-4 s London Mathematical Society monographs series 32 (DE-604)BV045355493 32 http://www.loc.gov/catdir/enhancements/fy0726/2006052879-d.html Publisher description Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016136158&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Davis, Michael The geometry and topology of Coxeter groups London Mathematical Society monographs series Coxeter groups Geometric group theory Algebraische Topologie (DE-588)4120861-4 gnd Geometrie (DE-588)4020236-7 gnd Coxeter-Gruppe (DE-588)4261522-7 gnd
subject_GND	(DE-588)4120861-4 (DE-588)4020236-7 (DE-588)4261522-7
title	The geometry and topology of Coxeter groups
title_auth	The geometry and topology of Coxeter groups
title_exact_search	The geometry and topology of Coxeter groups
title_exact_search_txtP	The geometry and topology of Coxeter groups
title_full	The geometry and topology of Coxeter groups Michael W. Davis
title_fullStr	The geometry and topology of Coxeter groups Michael W. Davis
title_full_unstemmed	The geometry and topology of Coxeter groups Michael W. Davis
title_short	The geometry and topology of Coxeter groups
title_sort	the geometry and topology of coxeter groups
topic	Coxeter groups Geometric group theory Algebraische Topologie (DE-588)4120861-4 gnd Geometrie (DE-588)4020236-7 gnd Coxeter-Gruppe (DE-588)4261522-7 gnd
topic_facet	Coxeter groups Geometric group theory Algebraische Topologie Geometrie Coxeter-Gruppe
url	http://www.loc.gov/catdir/enhancements/fy0726/2006052879-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016136158&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV045355493
work_keys_str_mv	AT davismichael thegeometryandtopologyofcoxetergroups

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