Leseprobe
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The Geometry and Topology of Coxeter Groups. (LMS-32)
Michael W. Davis
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
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.
Kundenbewertungen
Neighbourhood (mathematics), Dimension (vector space), Homology sphere, Algebraic K-theory, Fundamental polygon, Diagram (category theory), Intersection (set theory), Variable (mathematics), Category of abelian groups, Disk (mathematics), Coxeter group, Minor (linear algebra), Graph (discrete mathematics), Geometrization conjecture, Girth (graph theory), Topological manifold, Word problem (mathematics), Homotopy, Riemannian manifold, Torsor (algebraic geometry), Homology (mathematics), Theorem, Homotopy sphere, Topology, Hyperbolic 3-manifold, Von Neumann algebra, Convex polytope, Support (mathematics), Reflection group, Algebraic topology, Homotopy group, Projection (mathematics), Fundamental group, Polytope, Bounded set (topological vector space), K-cell (mathematics), Uniformization theorem, Topological space, Duality (mathematics), Geometric group theory, Commutative ring, Half-space (geometry), Compactification (mathematics), Subgroup, Algebraic group, Parity (mathematics), Hecke algebra, Hyperbolic manifold, Quotient space (topology), Isometry group, Degeneracy (mathematics), Three-dimensional space (mathematics), CW complex, Basis (linear algebra), Geometry, Simplicial complex, Lie algebra, Set (mathematics), Cayley graph, JSJ decomposition, Graph of groups, Characterization (mathematics), Fixed point (mathematics), Module (mathematics), Mathematical induction, Combinatorial group theory, Group algebra, Sphere theorem (3-manifolds), Mostow rigidity theorem, Connectivity (graph theory)