The Geometry And Topology of Coxeter Groups

by Davis, Michael W.

• ISBN13: 9780691131382
• ISBN10: 0691131384
• Format: Hardcover
• Copyright: 2007-10-29
• Publisher: Princeton Univ Pr

Summary

The Geometry and Topology of Coxeter Groupsis a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincareacute; Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.

Author Biography

Michael W. Davis is professor of mathematics at Ohio State University.

Table of Contents

Preface	p. xiii
Introduction and Preview	p. 1
Introduction	p. 1
A Preview of the Right-Angled Case	p. 9
Some Basic Notions in Geometric Group Theory	p. 15
Cayley Graphs and Word Metrics	p. 15
Cayley 2-Complexes	p. 18
Background on Aspherical Spaces	p. 21
Coxeter Groups	p. 26
Dihedral Groups	p. 26
Reflection Systems	p. 30
Coxeter Systems	p. 37
The Word Problem	p. 40
Coxeter Diagrams	p. 42
More Combinatorial Theory of Coxeter Groups	p. 44
Special Subgroups in Coxeter Groups	p. 44
Reflections	p. 46
The Shortest Element in a Special Coset	p. 47
Another Characterization of Coxeter Groups	p. 48
Convex Subsets of W	p. 49
The Element of Longest Length	p. 51
The Letters with Which a Reduced Expression Can End	p. 53
A Lemma of Tits	p. 55
Subgroups Generated by Reflections	p. 57
Normalizers of Special Subgroups	p. 59
The Basic Construction	p. 63
The Space u	p. 63
The Case of a Pre-Coxeter System	p. 66
Sectors in u	p. 68
Geometric Reflection Groups	p. 72
Linear Reflections	p. 73
Spaces of Constant Curvature	p. 73
Polytopes with Nonobutse Dihedral Angles	p. 78
The Developing Map	p. 81
Polygon Groups	p. 85
Finite Linear Groups Generated by Reflections	p. 87
Examples of Finite Reflection Groups	p. 92
Geometric Simplices: The Gram Matrix and the Cosine Matrix	p. 96
Simplicial Coxeter Groups: Lanner's Theorem	p. 102
Three-dimensional Hyperbolic Reflection Groups: Andreev's Theorem	p. 103
Higher-dimensional Hyperbolic Reflection Groups: Vinberg's Theorem	p. 110
The Canonical Representation	p. 115
The Complex [Sigma]	p. 123
The Nerve of a Coxeter System	p. 123
Geometric Realizations	p. 126
A Cell Structure on [Sigma]	p. 128
Examples	p. 132
Fixed Posets and Fixed Subspaces	p. 133
The Algebraic Topology of u and of [Sigma]	p. 136
The Homology of u	p. 137
Acyclicity Conditions	p. 140
Cohomology with Compact Supports	p. 146
The Case Where X Is a General Space	p. 150
Cohomology with Group Ring Coefficients	p. 152
Background on the Ends of a Group	p. 157
The Ends of W	p. 159
Splittings of Coxeter Groups	p. 160
Cohomology of Normalizers of Spherical Special Subgroups	p. 163
The Fundamental Group and the Fundamental Group at Infinity	p. 166
The Fundamental Group of u	p. 166
What Is [Sigma] Simply Connected at Infinity?	p. 170
Actions on Manifolds	p. 176
Reflection Groups on Manifolds	p. 177
The Tangent Bundle	p. 183
Background on Contractible Manifolds	p. 185
Background on Homology Manifolds	p. 191
Aspherical Manifolds Not Covered by Euclidean Space	p. 195
When Is [Sigma] a Manifold?	p. 197
Reflection Groups on Homology Manifolds	p. 197
Generalized Homology Spheres and Polytopes	p. 201
Virtual Poincare Duality Groups	p. 205
The Reflection Group Trick	p. 212
The First Version of the Trick	p. 212
Examples of Fundamental Groups of Closed Aspherical Manifolds	p. 215
Nonsmoothable Aspherical Manifolds	p. 216
The Borel Conjecture and the PD[superscript n]-Group Conjecture	p. 217
The Second Version of the Trick	p. 220
The Bestvina-Brady Examples	p. 222
The Equivariant Reflection Group Trick	p. 225
[Sigma] is Cat(O): Theorems of Gromov and Moussong	p. 230
A Piecewise Euclidean Cell Structure on [Sigma]	p. 231
The Right-Angled Case	p. 233
The General Case	p. 234
The Visual Boundary of [Sigma]	p. 237
Background on Word Hyperbolic Groups	p. 238
When Is [Sigma] CAT(-1)?	p. 241
Free Abelian Subgroups of Coxeter Groups	p. 245
Relative Hyperbolization	p. 247
Rigidity	p. 255
Definitions, Examples, Counterexamples	p. 255
Spherical Parabolic Subgroups and Their Fixed Subspaces	p. 260
Coxeter Groups of Type PM	p. 263
Strong Rigidity for Groups of Type PM	p. 268
Free Quotients and Surface Subgroups	p. 276
Largeness	p. 276
Surface Subgroups	p. 282
Another Look at (Co)Homology	p. 286
Cohomology with Constant Coefficients	p. 286
Decompositions of Coefficient Systems	p. 288
The W-Module Structure on (Co)homology	p. 295
The Case Where W Is finite	p. 303
The Euler Characteristic	p. 306
Background on Euler Characteristics	p. 306
The Euler Characteristic Conjecture	p. 310
The Flag Complex Conjecture	p. 313
Growth Series	p. 315
Rationality of the Growth Series	p. 315
Exponential versus Polynomial Growth	p. 322
Reciprocity	p. 324
Relationship with the h-Polynomial	p. 325
Buildings	p. 328
The Combinatorial Theory of Buildings	p. 328
The Geometric Realization of a Building	p. 336
Buildings Are CAT(0)	p. 338
Euler-Poincare Measure	p. 341
Hecke-Von Neumann Algebras	p. 344
Hecke Algebras	p. 344
Hecke-Von Neumann Algebras	p. 349
Weighted L[superscript 2]-(Co)homology	p. 359
Weighted L[superscript 2]-(Co)homology	p. 361
Weighted L[superscript 2]-Betti Numbers and Euler Characteristics	p. 366
Concentration of (Co)homology in Dimension 0	p. 368
Weighted Poincare Duality	p. 370
A Weighted Version of the Singer Conjecture	p. 374
Decomposition Theorems	p. 376
Decoupling Cohomology	p. 389
L[superscript 2]-Cohomology of Buildings	p. 394
Cell Complexes	p. 401
Cells and Cell Complexes	p. 401
Posets and Abstract Simplicial Complexes	p. 406
Flag Complexes and Barycentric Subdivisions	p. 409
Joins	p. 412
Faces and Cofaces	p. 415
Links	p. 418
Regular Polytopes	p. 421
Chambers in the Barycentric Subdivision of a Polytope	p. 421
Classification of Regular Polytopes	p. 424
Regular Tessellations of Spheres	p. 426
Regular Tessellations	p. 428
The Classification of Spherical and Euclidean Coxeter Groups	p. 433
Statements of the Classification Theorems	p. 433
Calculating Some Determinants	p. 434
Proofs of the Classification Theorems	p. 436
The Geometric Representation	p. 439
Injectivity of the Geometric Representation	p. 439
The Tits Cone	p. 442
Complement on Root Systems	p. 446
Complexes of Groups	p. 449
Background on Graphs of Groups	p. 450
Complexes of Groups	p. 454
The Meyer-Vietoris Spectral Sequence	p. 459
Homology and Cohomology of Groups	p. 465
Some Basic Definitions	p. 465
Equivalent (Co)homology with Group Ring Coefficients	p. 467
Cohomological Dimension and Geometric Dimension	p. 470
Finiteness Conditions	p. 471
Poincare Duality Groups and Duality Groups	p. 474
Algebraic Topology at Infinity	p. 477
Some Algebra	p. 477
Homology and Cohomology at Infinity	p. 479
Ends of a Space	p. 482
Semistability and the Fundamental Group at Infinity	p. 483
The Novikov and Borel Conjectures	p. 487
Around the Borel Conjecture	p. 487
Smoothing Theory	p. 491
The Surgery Exact Sequence and the Assembly Map Conjecture	p. 493
The Novikov Conjecture	p. 496
Nonpositive Curvature	p. 499
Geodesic Metric Spaces	p. 499
The CAT([kappa])-Inequality	p. 499
Polyhedra of Piecewise Constant Curvature	p. 507
Properties of CAT(0) Groups	p. 511
Piecewise Spherical Polyhedra	p. 513
Gromov's Lemma	p. 516
Moussong's Lemma	p. 520
The Visual Boundary of a CAT(0)-Space	p. 524
L[superscript 2]-(Co)Homology	p. 531
Background on von Neuman Algebras	p. 531
The Regular Representation	p. 531
L[superscript 2]-(Co)homology	p. 538
Basic L[superscript 2] Algebraic Topology	p. 541
L[superscript 2]-Betti Numbers and Euler Characteristics	p. 544
Poincare Duality	p. 546
The Singer Conjecture	p. 547
Vanishing Theorems	p. 548
Bibliography	p. 555
Index	p. 573
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