The Geometry And Topology Of Coxeter Groups

Author : Michael Davis
Publisher : Princeton University Press
Page : 601 pages
File Size : 41,5 Mb
Release : 2008
Category : Mathematics
ISBN : 9780691131382

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The Geometry and Topology of Coxeter Groups by Michael Davis Pdf

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.

Author : Michael Davis
Publisher : Princeton University Press
Page : 600 pages
File Size : 52,6 Mb
Release : 2012-11-26
Category : Mathematics
ISBN : 9781400845941

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The Geometry and Topology of Coxeter Groups. (LMS-32) by Michael Davis Pdf

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.

Author : Anne Thomas
Publisher : Unknown
Page : 128 pages
File Size : 44,5 Mb
Release : 2018
Category : MATHEMATICS
ISBN : 3037196890

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Geometric and Topological Aspects of Coxeter Groups and Buildings by Anne Thomas Pdf

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.

Author : Anne Thomas
Publisher : Unknown
Page : 128 pages
File Size : 44,5 Mb
Release : 2018
Category : Electronic
ISBN : 3037191899

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Geometric and Topological Aspects of Coxeter Groups and Buildings by Anne Thomas Pdf

Author : Matt Clay,Dan Margalit
Publisher : Princeton University Press
Page : 456 pages
File Size : 41,8 Mb
Release : 2017-07-11
Category : Mathematics
ISBN : 9781400885398

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Office Hours with a Geometric Group Theorist by Matt Clay,Dan Margalit Pdf

Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. Office Hours with a Geometric Group Theorist brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It's like having office hours with your most trusted math professors. An essential primer for undergraduates making the leap to graduate work, the book begins with free groups—actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson's groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples. Accessible to students who have taken a first course in abstract algebra, Office Hours with a Geometric Group Theorist also features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.

Author : Anders Bjorner,Francesco Brenti
Publisher : Springer Science & Business Media
Page : 371 pages
File Size : 53,5 Mb
Release : 2006-02-25
Category : Mathematics
ISBN : 9783540275961

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Combinatorics of Coxeter Groups by Anders Bjorner,Francesco Brenti Pdf

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

Author : James E. Humphreys
Publisher : Cambridge University Press
Page : 222 pages
File Size : 51,9 Mb
Release : 1992-10
Category : Mathematics
ISBN : 0521436133

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Reflection Groups and Coxeter Groups by James E. Humphreys Pdf

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.

Author : Alexandre V. Borovik,Israel M. Gelfand,Neil White
Publisher : Springer Science & Business Media
Page : 282 pages
File Size : 40,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461220664

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Coxeter Matroids by Alexandre V. Borovik,Israel M. Gelfand,Neil White Pdf

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.

Author : Howard Hiller
Publisher : Pitman Publishing
Page : 230 pages
File Size : 50,7 Mb
Release : 1982
Category : Mathematics
ISBN : UOM:39015049314548

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Geometry of Coxeter Groups by Howard Hiller Pdf

Author : Richard Kane
Publisher : Springer Science & Business Media
Page : 382 pages
File Size : 45,7 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475735420

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Reflection Groups and Invariant Theory by Richard Kane Pdf

Reflection groups and invariant theory is a branch of mathematics that lies at the intersection between geometry and algebra. The book contains a deep and elegant theory, evolved from various graduate courses given by the author over the past 10 years.

Author : Craig D. Hodgson,William H. Jaco,Martin G. Scharlemann,Stephan Tillmann
Publisher : American Mathematical Soc.
Page : 395 pages
File Size : 42,5 Mb
Release : 2013-08-23
Category : Mathematics
ISBN : 9780821884805

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Geometry and Topology Down Under by Craig D. Hodgson,William H. Jaco,Martin G. Scharlemann,Stephan Tillmann Pdf

This book contains the proceedings of the conference Geometry & Topology Down Under, held July 11-22, 2011, at the University of Melbourne, Parkville, Australia, in honour of Hyam Rubinstein. The main topic of the book is low-dimensional geometry and topology. It includes both survey articles based on courses presented at the conferences and research articles devoted to important questions in low-dimensional geometry. Together, these contributions show how methods from different fields of mathematics contribute to the study of 3-manifolds and Gromov hyperbolic groups. It also contains a list of favorite problems by Hyam Rubinstein.

Author : Victor M. Buchstaber,Taras E. Pano
Publisher : American Mathematical Soc.
Page : 518 pages
File Size : 49,5 Mb
Release : 2015-07-15
Category : Algebraic topology
ISBN : 9781470422141

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Toric Topology by Victor M. Buchstaber,Taras E. Pano Pdf

This book is about toric topology, a new area of mathematics that emerged at the end of the 1990s on the border of equivariant topology, algebraic and symplectic geometry, combinatorics, and commutative algebra. It has quickly grown into a very active area with many links to other areas of mathematics, and continues to attract experts from different fields. The key players in toric topology are moment-angle manifolds, a class of manifolds with torus actions defined in combinatorial terms. Construction of moment-angle manifolds relates to combinatorial geometry and algebraic geometry of toric varieties via the notion of a quasitoric manifold. Discovery of remarkable geometric structures on moment-angle manifolds led to important connections with classical and modern areas of symplectic, Lagrangian, and non-Kaehler complex geometry. A related categorical construction of moment-angle complexes and polyhedral products provides for a universal framework for many fundamental constructions of homotopical topology. The study of polyhedral products is now evolving into a separate subject of homotopy theory. A new perspective on torus actions has also contributed to the development of classical areas of algebraic topology, such as complex cobordism. This book includes many open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter this beautiful new area.

Author : Anonim
Publisher : Unknown
Page : 654 pages
File Size : 55,6 Mb
Release : 2008
Category : Geometry
ISBN : UOM:39015072617122

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Geometry & Topology by Anonim Pdf

Fully refereed international journal dealing with all aspects of geometry and topology and their applications.

Author : Marcelo Aguiar
Publisher : American Mathematical Soc.
Page : 201 pages
File Size : 42,6 Mb
Release : 2006
Category : Education
ISBN : 9780821853542

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Coxeter Groups and Hopf Algebras by Marcelo Aguiar Pdf

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.

Author : Siobhan Roberts
Publisher : Unknown
Page : 400 pages
File Size : 46,8 Mb
Release : 2007-07
Category : Geometry
ISBN : 0887847773

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King of Infinite Space by Siobhan Roberts Pdf

In the tradition of A Beautiful Mind, here is an engaging biography of a world-famous Canadian geometry genius, Donald Coxeter, whose work has slowly and steadily become the air we breathe, with everyday applications in telecommunication and e-commerce, among other fields. King of Infinite Space combines two bestselling categories, popular science and biography. It makes geometry accessible through the fascinating life of the 20th century's greatest classical geometer -- a man who also happens to be a Canadian. Coxeter's story weaves together several rich and surprising threads, from the idiosyncratic nature of his personal life to the geopolitical fabric of his times -- what he calls the wild wicked world -- and illuminates the importance of mathematics therein. Indeed, Coxeter is credited with saving classical geometry from near extinction: for over fifty years, he fought a quiet war of attrition against disdain for his unfashionable geometry, stoically retaining his passion for the beauty of shapes.