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Publisher:

Princeton University Press

Publication Date:

2007

Number of Pages:

600

Format:

Hardcover

Series:

London Mathematical Society Monographs 32

Price:

85.00

ISBN:

9780691131382

Category:

Monograph

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

Gizem Karaali

04/30/2009

This is not really an e-book, but comes quite close. The full text is available on its author's web page along with a list of errata and even the MathSciNet review. One might expect that this could initially make it less likely for some to purchase the hard copy, especially graduate students living on a tight budget. However, if the author's intention is to be read, then this is a very good tactic. Those interested in Coxeter groups will have the opportunity to take a serious look into the book and, if my hunch is correct, will be quite excited by what they see. They may even get so thrilled that they will go ahead and buy the beautiful hardcover. I think more mathematician-authors should consider making their manuscripts (at least partially) available online; I salute Michael Davis for doing it (and his publisher for going along with it).

Now to the book itself.

Coxeter groups are ubiquitous in modern mathematics. There are therefore several books on the subject in various flavors catering to readers with differing priorities. A representation theorist's choice will most likely be Humphreys' *Reflection Groups and Coxeter Groups*, while a combinatorialist will probably go for Björner and Brenti's Combinatorics of Coxeter Groups. Michael Davis has written the one appropriate for geometric group theorists (finally!). Though this is undeniably a research monograph that focuses mainly on the geometric group theory point of view, I can easily see the first five chapters of it being a useful introduction for anyone interested in Coxeter groups.

The first chapter is an introduction to the whole text, and therefore can be skimmed through at a first reading, especially by readers who are looking to gain an insight into the main focus of the text but are not yet ready to delve into the details. Chapter 2 is a sufficiently comprehensive exposition of the basic ideas of geometric group theory that will be necessary in the rest of the book. The next two chapters provide the foundational definitions, examples and (combinatorial) constructions necessary for any study of Coxeter groups, and Chapter 5 introduces the basic geometric realization of a given Coxeter system. Thus the reader who comes to Chapter 6 has already been exposed to both the combinatorial and geometric approaches.

This can be a good stopping point for those who were merely looking for an introduction to Coxeter groups. Reading only this far would still be a good use of one's time. However for those who want to learn more about geometric group theory and the role Coxeter groups play in it, leading up to and including most recent research in the area, the rest of the book will also prove to be a well-chosen read. The fact that at that point the text begins to read more like a research monograph will definitely be acceptable to such a reader. The author has personally contributed to several of the advanced ideas and techniques developed in these latter chapters, and he does a great job bringing all of the relevant strands together in this volume.

Readers who want to know more about the contents should definitely look into Ralf Gramlich's excellent MathSciNet review, which as you already read above is available on the author's web page. I will simply add a comment on the appendices: The book comes with ten appendices which make up almost one third of the book. These vary in difficulty but any reader will find at least one of them very useful. The topics covered in the appendices range from background material (e.g., cell complexes in Appendix A, regular polytopes in Appendix B, complexes of groups in Appendix E, homology and cohomology of groups in Appendix F) to material that extends what is already covered in the main body of the text (eg. the classification of spherical and Euclidean Coxeter groups in Appendix C, the geometric representation of a Coxeter group in Appendix D). Overall these appendices are an invaluable component of this book, and make it even more useful for students.

Gizem Karaali is assistant professor of Mathematics at Pomona College.

Preface xiii

Chapter 1: INTRODUCTION AND PREVIEW 1

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 15

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

4.10 Normalizers of Special Subgroups 59

Chapter 5: THE BASIC CONSTRUCTION 63

5.1 The Space *U* 63

5.2 The Case of a Pre-Coxeter System 66

5.3 Sectors in *U* 68

Chapter 6: 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 Simplices: The Gram Matrix and the Cosine Matrix 96

6.9 Simplicial Coxeter Groups: Lann´er'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 ∑ 123

7.1 The Nerve of a Coxeter 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 133

Chapter 8: THE ALGEBRAIC TOPOLOGY OF *U* AND OF ∑ 136

8.1 The Homology of *U* 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 157

8.7 The Ends of *W* 159

8.8 Splittings of Coxeter Groups 160

8.9 Cohomology of Normalizers of Spherical Special Subgroups 163

Chapter 9: THE FUNDAMENTAL GROUP AND THE FUNDAMENTAL

GROUP AT INFINITY 166

9.1 The Fundamental Group of *U* 166

9.2 What Is ∑ Simply Connected at Infinity? 170

Chapter 10: ACTIONS ON MANIFOLDS 176

10.1 Reflection Groups on Manifolds 177

10.2 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 ∑ a Manifold? 197

10.7 Reflection Groups on Homology Manifolds 197

10.8 Generalized Homology Spheres and Polytopes 201

10.9 Virtual Poincar´e Duality Groups 205

Chapter 11: THE REFLECTION GROUP TRICK 212

11.1 The First Version of the Trick 212

11.2 Examples of Fundamental Groups of Closed Aspherical

Manifolds 215

11.3 Nonsmoothable Aspherical Manifolds 216

11.4 The Borel Conjecture and the PDn-Group Conjecture 217

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 12: ∑ IS CAT(0): THEOREMS OF GROMOV 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 The Visual Boundary of ∑ 237

12.5 Background on Word Hyperbolic Groups 238

12.6 When Is ∑ CAT(-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

Chapter 15: ANOTHER LOOK AT (CO)HOMOLOGY 286

15.1 Cohomology with Constant Coefficients 286

15.2 Decompositions of Coefficient Systems 288

15.3 The W-Module Structure on (Co)homology 295

15.4 The Case Where W Is finite 303

Chapter 16: 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

17.1 Rationality of the Growth Series 315

17.2 Exponential versus Polynomial Growth 322

17.3 Reciprocity 324

17.4 Relationship with the *h*-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(0) 338

18.4 Euler-Poincar´e Measure 341

Chapter 19: HECKE-VON NEUMANN ALGEBRAS 344

19.1 Hecke Algebras 344

19.2 Hecke-Von Neumann Algebras 349

Chapter 20: WEIGHTED *L*^{2}-(CO)HOMOLOGY 359

20.1 Weighted *L*^{2}-(Co)homology 361

20.2 Weighted *L*^{2}-Betti Numbers and Euler Characteristics 366

20.3 Concentration of (Co)homology in Dimension 0 368

20.4 Weighted Poincar´e Duality 370

20.5 A Weighted Version of the Singer Conjecture 374

20.6 Decomposition Theorems 376

20.7 Decoupling Cohomology 389

20.8 *L*^{2}-Cohomology of Buildings 394

Appendix A: CELL COMPLEXES 401

A.1 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

A.5 Faces and Cofaces 415

A.6 Links 418

Appendix B: REGULAR POLYTOPES 421

B.1 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 C: THE CLASSIFICATION OF SPHERICAL AND EUCLIDEAN COXETER GROUPS 433

C.1 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 Poincar´e Duality Groups and Duality Groups 474

Appendix G: ALGEBRAIC TOPOLOGY AT INFINITY 477

G.1 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.1 Around the Borel 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

Appendix I: NONPOSITIVE CURVATURE 499

I.1 Geodesic Metric Spaces 499

I.2 The CAT(?)-Inequality 499

I.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 516

I.7 Moussong's Lemma 520

I.8 The Visual Boundary of a CAT(0)-Space 524

Appendix J: *L*^{2}-(CO)HOMOLOGY 531

J.1 Background on von Neumann Algebras 531

J.2 The Regular Representation 531

J.3 *L*^{2}-(Co)homology 538

J.4 Basic *L*^{2} Algebraic Topology 541

J.5 *L*^{2}-Betti Numbers and Euler Characteristics 544

J.6 Poincar´e Duality 546

J.7 The Singer Conjecture 547

J.8 Vanishing Theorems 548

Bibliography 555

Index 573

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