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The Geometry and Topology of Coxeter Groups. (LMS-32).
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...
| Clasificación: | Libro Electrónico |
|---|---|
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Princeton University Press
2012.
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| Colección: | London Mathematical Society monographs.
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| Temas: | |
| Acceso en línea: | Texto completo |
MARC
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| 001 | EBOOKCENTRAL_ocn823283891 | ||
| 003 | OCoLC | ||
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| 008 | 121202s2012 xx o 000 0 eng d | ||
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| 245 | 0 | 4 | |a The Geometry and Topology of Coxeter Groups. (LMS-32). |
| 260 | |b Princeton University Press |c 2012. | ||
| 300 | |a 1 online resource (601 pages) | ||
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| 490 | 1 | |a London Mathematical Society Monographs | |
| 520 | |a 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. | ||
| 505 | 0 | |a Cover; Contents; Preface; Chapter 1 INTRODUCTION AND PREVIEW; 1.1 Introduction; 1.2 A Preview of the Right-Angled Case; Chapter 2 SOME BASIC NOTIONS IN GEOMETRIC GROUP THEORY; 2.1 Cayley Graphs and Word Metrics; 2.2 Cayley 2-Complexes; 2.3 Background on Aspherical Spaces; Chapter 3 COXETER GROUPS; 3.1 Dihedral Groups; 3.2 Reflection Systems; 3.3 Coxeter Systems; 3.4 The Word Problem; 3.5 Coxeter Diagrams; Chapter 4 MORE COMBINATORIAL THEORY OF COXETER GROUPS; 4.1 Special Subgroups in Coxeter Groups; 4.2 Reflections; 4.3 The Shortest Element in a Special Coset | |
| 505 | 8 | |a 4.4 Another Characterization of Coxeter Groups4.5 Convex Subsets of W; 4.6 The Element of Longest Length; 4.7 The Letters with Which a Reduced Expression Can End; 4.8 A Lemma of Tits; 4.9 Subgroups Generated by Reflections; 4.10 Normalizers of Special Subgroups; Chapter 5 THE BASIC CONSTRUCTION; 5.1 The Space U; 5.2 The Case of a Pre-Coxeter System; 5.3 Sectors in U; Chapter 6 GEOMETRIC REFLECTION GROUPS; 6.1 Linear Reflections; 6.2 Spaces of Constant Curvature; 6.3 Polytopes with Nonobtuse Dihedral Angles; 6.4 The Developing Map; 6.5 Polygon Groups | |
| 590 | |a ProQuest Ebook Central |b Ebook Central Academic Complete | ||
| 650 | 0 | |a Coxeter groups. | |
| 650 | 6 | |a Groupes de Coxeter. | |
| 650 | 7 | |a Coxeter groups |2 fast | |
| 720 | |a Davis, Michael W. | ||
| 776 | 0 | 8 | |i Print version: |a Davis, Michael W. |t Geometry and Topology of Coxeter Groups. (LMS-32). |d Princeton : Princeton University Press, ©2012 |z 9780691131382 |
| 830 | 0 | |a London Mathematical Society monographs. | |
| 856 | 4 | 0 | |u https://ebookcentral.uam.elogim.com/lib/uam-ebooks/detail.action?docID=1062357 |z Texto completo |
| 880 | 8 | |6 505-00/(S |a C.2 Calculating Some Determinants -- C.3 Proofs of the Classification Theorems -- Appendix D: THE GEOMETRIC REPRESENTATION -- D.1 Injectivity of the Geometric Representation -- D.2 The Tits Cone -- D.3 Complement on Root Systems -- Appendix E: COMPLEXES OF GROUPS -- E.1 Background on Graphs of Groups -- E.2 Complexes of Groups -- E.3 The Meyer-Vietoris Spectral Sequence -- Appendix F: HOMOLOGY AND COHOMOLOGY OF GROUPS -- F.1 Some Basic Definitions -- F.2 Equivalent (Co)homology with Group Ring Coefficients -- F.3 Cohomological Dimension and Geometric Dimension -- F.4 Finiteness Conditions -- F.5 Poincaré Duality Groups and Duality Groups -- Appendix G: ALGEBRAIC TOPOLOGY AT INFINITY -- G.1 Some Algebra -- G.2 Homology and Cohomology at Infinity -- G.3 Ends of a Space -- G.4 Semistability and the Fundamental Group at Infinity -- Appendix H: THE NOVIKOV AND BOREL CONJECTURES -- H.1 Around the Borel Conjecture -- H.2 Smoothing Theory -- H.3 The Surgery Exact Sequence and the Assembly Map Conjecture -- H.4 The Novikov Conjecture -- Appendix I: NONPOSITIVE CURVATURE -- I.1 Geodesic Metric Spaces -- I.2 The CAT(κ)-Inequality -- I.3 Polyhedra of Piecewise Constant Curvature -- I.4 Properties of CAT(0) Groups -- I.5 Piecewise Spherical Polyhedra -- I.6 Gromov's Lemma -- I.7 Moussong's Lemma -- I.8 The Visual Boundary of a CAT(0)-Space -- Appendix J: L[sup(2)]-(CO)HOMOLOGY -- J.1 Background on von Neumann Algebras -- J.2 The Regular Representation -- J.3 L[sup(2)]-(Co)homology -- J.4 Basic L[sup(2)] Algebraic Topology -- J.5 L[sup(2)]-Betti Numbers and Euler Characteristics -- J.6 Poincaré Duality -- J.7 The Singer Conjecture -- J.8 Vanishing Theorems -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- X -- Z. | |
| 880 | 0 | |6 505-00/(S |a Cover -- Contents -- Preface -- Chapter 1 INTRODUCTION AND PREVIEW -- 1.1 Introduction -- 1.2 A Preview of the Right-Angled Case -- Chapter 2 SOME BASIC NOTIONS IN GEOMETRIC GROUP THEORY -- 2.1 Cayley Graphs and Word Metrics -- 2.2 Cayley 2-Complexes -- 2.3 Background on Aspherical Spaces -- Chapter 3 COXETER GROUPS -- 3.1 Dihedral Groups -- 3.2 Reflection Systems -- 3.3 Coxeter Systems -- 3.4 The Word Problem -- 3.5 Coxeter Diagrams -- Chapter 4 MORE COMBINATORIAL THEORY OF COXETER GROUPS -- 4.1 Special Subgroups in Coxeter Groups -- 4.2 Reflections -- 4.3 The Shortest Element in a Special Coset -- 4.4 Another Characterization of Coxeter Groups -- 4.5 Convex Subsets of W -- 4.6 The Element of Longest Length -- 4.7 The Letters with Which a Reduced Expression Can End -- 4.8 A Lemma of Tits -- 4.9 Subgroups Generated by Reflections -- 4.10 Normalizers of Special Subgroups -- Chapter 5 THE BASIC CONSTRUCTION -- 5.1 The Space U -- 5.2 The Case of a Pre-Coxeter System -- 5.3 Sectors in U -- Chapter 6 GEOMETRIC REFLECTION GROUPS -- 6.1 Linear Reflections -- 6.2 Spaces of Constant Curvature -- 6.3 Polytopes with Nonobtuse Dihedral Angles -- 6.4 The Developing Map -- 6.5 Polygon Groups -- 6.6 Finite Linear Groups Generated by Reflections -- 6.7 Examples of Finite Reflection Groups -- 6.8 Geometric Simplices: The Gram Matrix and the Cosine Matrix -- 6.9 Simplicial Coxeter Groups: Lannér's Theorem -- 6.10 Three-dimensional Hyperbolic Reflection Groups: Andreev's Theorem -- 6.11 Higher-dimensional Hyperbolic Reflection Groups: Vinberg's Theorem -- 6.12 The Canonical Representation -- Chapter 7 THE COMPLEX Σ -- 7.1 The Nerve of a Coxeter System -- 7.2 Geometric Realizations -- 7.3 A Cell Structure on Σ -- 7.4 Examples -- 7.5 Fixed Posets and Fixed Subspaces -- Chapter 8 THE ALGEBRAIC TOPOLOGY OF U AND OF Σ -- 8.1 The Homology of U. | |
| 880 | 8 | |6 505-00/(S |a 8.2 Acyclicity Conditions -- 8.3 Cohomology with Compact Supports -- 8.4 The Case Where X Is a General Space -- 8.5 Cohomology with Group Ring Coefficients -- 8.6 Background on the Ends of a Group -- 8.7 The Ends of W -- 8.8 Splittings of Coxeter Groups -- 8.9 Cohomology of Normalizers of Spherical Special Subgroups -- Chapter 9 THE FUNDAMENTAL GROUP AND THE FUNDAMENTAL GROUP AT INFINITY -- 9.1 The Fundamental Group of U -- 9.2 What Is Σ Simply Connected at Infinity-- Chapter 10 ACTIONS ON MANIFOLDS -- 10.1 Reflection Groups on Manifolds -- 10.2 The Tangent Bundle -- 10.3 Background on Contractible Manifolds -- 10.4 Background on Homology Manifolds -- 10.5 Aspherical Manifolds Not Covered by Euclidean Space -- 10.6 When Is Σ a Manifold-- 10.7 Reflection Groups on Homology Manifolds -- 10.8 Generalized Homology Spheres and Polytopes -- 10.9 Virtual Poincaré Duality Groups -- Chapter 11 THE REFLECTION GROUP TRICK -- 11.1 The First Version of the Trick -- 11.2 Examples of Fundamental Groups of Closed Aspherical Manifolds -- 11.3 Nonsmoothable Aspherical Manifolds -- 11.4 The Borel Conjecture and the PD[sup(n)]-Group Conjecture -- 11.5 The Second Version of the Trick -- 11.6 The Bestvina-Brady Examples -- 11.7 The Equivariant Reflection Group Trick -- Chapter 12 Σ IS CAT(O): THEOREMS OF GROMOV AND MOUSSONG -- 12.1 A Piecewise Euclidean Cell Structure on Σ -- 12.2 The Right-Angled Case -- 12.3 The General Case -- 12.4 The Visual Boundary of Σ -- 12.5 Background on Word Hyperbolic Groups -- 12.6 When Is Σ CAT( -1)-- 12.7 Free Abelian Subgroups of Coxeter Groups -- 12.8 Relative Hyperbolization -- Chapter 13 RIGIDITY -- 13.1 Definitions, Examples, Counterexamples -- 13.2 Spherical Parabolic Subgroups and Their Fixed Subspaces -- 13.3 Coxeter Groups of Type PM -- 13.4 Strong Rigidity for Groups of Type PM. | |
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