The Geometry And Topology Of Coxeter Groups
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The Geometry And Topology Of Coxeter Groups
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Author : Michael Davis
language : en
Publisher: Princeton University Press
Release Date : 2008
The Geometry And Topology Of Coxeter Groups written by Michael Davis and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.
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.
The Geometry And Topology Of Coxeter Groups Lms 32
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Author : Michael W. Davis
language : en
Publisher: Princeton University Press
Release Date : 2012-11-26
The Geometry And Topology Of Coxeter Groups Lms 32 written by Michael W. Davis and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-11-26 with Mathematics categories.
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.
The Geometry And Topology Of Coxeter Groups
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Author : Michael W. Davis
language : en
Publisher: Springer
Release Date : 2025-07-03
The Geometry And Topology Of Coxeter Groups written by Michael W. Davis and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-07-03 with Mathematics categories.
This book, now in a revised and extended second edition, offers an in-depth account of Coxeter groups through the perspective of geometric group theory. It examines the connections between Coxeter groups and major open problems in topology related to aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer Conjectures. The book also discusses key 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. In addition, this second edition includes new chapters on Artin groups and their Betti numbers. Written by a leading expert, the book is an authoritative reference on the subject.
Combinatorics Of Coxeter Groups
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Author : Anders Bjorner
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-02-25
Combinatorics Of Coxeter Groups written by Anders Bjorner and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-02-25 with Mathematics categories.
Includes a rich variety of exercises to accompany the exposition of Coxeter groups Coxeter groups have already been exposited from algebraic and geometric perspectives, but this book will be presenting the combinatorial aspects of Coxeter groups
Reflection Groups And Coxeter Groups
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Author : James E. Humphreys
language : en
Publisher: Cambridge University Press
Release Date : 1992-10
Reflection Groups And Coxeter Groups written by James E. Humphreys and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992-10 with Mathematics categories.
This graduate textbook presents a concrete and up-to-date introduction to the theory of Coxeter groups. The book is self-contained, making it suitable either for courses and seminars or for self-study. The first part is devoted to establishing concrete examples. Finite reflection groups acting on Euclidean spaces are discussed, and the first part ends with the construction of the affine Weyl groups, a class of Coxeter groups that plays a major role in Lie theory. The second part (which is logically independent of, but motivated by, the first) develops from scratch the properties of Coxeter groups in general, including the Bruhat ordering and the seminal work of Kazhdan and Lusztig on representations of Hecke algebras associated with Coxeter groups is introduced. Finally a number of interesting complementary topics as well as connections with Lie theory are sketched. The book concludes with an extensive bibliography on Coxeter groups and their applications.
Geometry Of Coxeter Groups
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Author : Howard Hiller
language : en
Publisher: Pitman Publishing
Release Date : 1982
Geometry Of Coxeter Groups written by Howard Hiller and has been published by Pitman Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 1982 with Mathematics categories.
Coxeter Groups And Hopf Algebras
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Author : Marcelo Aguiar
language : en
Publisher: American Mathematical Soc.
Release Date : 2006
Coxeter Groups And Hopf Algebras written by Marcelo Aguiar and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Education categories.
An important idea in the work of G.-C. Rota is that certain combinatorial objects give rise to Hopf algebras that reflect the manner in which these objects compose and decompose. Recent work has seen the emergence of several interesting Hopf algebras of this kind, which connect diverse subjects such as combinatorics, algebra, geometry, and theoretical physics. This monograph presents a novel geometric approach using Coxeter complexes and the projection maps of Tits for constructing and studying many of these objects as well as new ones. The first three chapters introduce the necessary background ideas making this work accessible to advanced graduate students. The later chapters culminate in a unified and conceptual construction of several Hopf algebras based on combinatorial objects which emerge naturally from the geometric viewpoint. This work lays a foundation and provides new insights for further development of the subject.
Coxeter Matroids
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Author : Alexandre V. Borovik
language : en
Publisher: Birkhäuser
Release Date : 2003-07-11
Coxeter Matroids written by Alexandre V. Borovik and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-07-11 with Mathematics categories.
Matroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry, and "Coxeter Matroids" provides an intuitive and interdisciplinary treatment of their theory. In this text, matroids are examined in terms of symmetric and finite reflection groups; also, symplectic matroids and the more general coxeter matroids are carefully developed. The Gelfand-Serganova theorem, which allows for the geometric interpretation of matroids as convex polytopes with certain symmetry properties, is presented, and in the final chapter, matroid representations and combinatorial flag varieties are discussed. With its excellent bibliography and index and ample references to current research, this work will be useful for graduate students and research mathematicians.
Geometric And Topological Aspects Of Coxeter Groups And Buildings
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Author : Anne Thomas
language : en
Publisher:
Release Date : 2018
Geometric And Topological Aspects Of Coxeter Groups And Buildings written by Anne Thomas and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with MATHEMATICS categories.
Coxeter groups are groups generated by reflections, and they appear throughout mathematics. Tits developed the general theory of Coxeter groups in order to develop the theory of buildings. Buildings have interrelated algebraic, combinatorial and geometric structures, and are powerful tools for understanding the groups which act on them. These notes focus on the geometry and topology of Coxeter groups and buildings, especially nonspherical cases. The emphasis is on geometric intuition, and there are many examples and illustrations. Part I describes Coxeter groups and their geometric realisations, particularly the Davis complex, and Part II gives a concise introduction to buildings. This book will be suitable for mathematics graduate students and researchers in geometric group theory, as well as algebra and combinatorics. The assumed background is basic group theory, including group actions, and basic algebraic topology, together with some knowledge of Riemannian geometry.