The geometry and topology of Coxeter groups

Michael W. Davis0691131384, 9780691131382

The Geometry and Topology of Coxeter Groups is 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 Poincaré 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.

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Table of contents :
prelims.pdf……Page 1
chapter1.pdf……Page 17
chapter2.pdf……Page 31
chapter3.pdf……Page 42
chapter4.pdf……Page 60
chapter5.pdf……Page 79
chapter6.pdf……Page 88
chapter7.pdf……Page 139
chapter8.pdf……Page 152
chapter9.pdf……Page 182
chapter10.pdf……Page 192
chapter11.pdf……Page 228
chapter12.pdf……Page 246
chapter13.pdf……Page 271
chapter14.pdf……Page 292
chapter15.pdf……Page 302
chapter16.pdf……Page 322
chapter17.pdf……Page 331
chapter18.pdf……Page 344
chapter19.pdf……Page 360
chapter20.pdf……Page 375
appendixa.pdf……Page 417
appendixb.pdf……Page 437
appendixc.pdf……Page 449
appendixd.pdf……Page 455
appendixe.pdf……Page 465
appendixf.pdf……Page 481
appendixg.pdf……Page 493
appendixh.pdf……Page 503
appendixi.pdf……Page 515
appendixj.pdf……Page 547
references.pdf……Page 571
index.pdf……Page 589