Geometric and Topological Aspects of Coxeter Groups and Buildings
Title | Geometric and Topological Aspects of Coxeter Groups and Buildings PDF eBook |
Author | Anne Thomas |
Publisher | |
Pages | |
Release | 2018 |
Genre | MATHEMATICS |
ISBN | 9783037196892 |
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.
The Geometry and Topology of Coxeter Groups
Title | The Geometry and Topology of Coxeter Groups PDF eBook |
Author | Michael Davis |
Publisher | Princeton University Press |
Pages | 601 |
Release | 2008 |
Genre | Mathematics |
ISBN | 0691131384 |
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.
Geometric and Topological Aspects of Coxeter Groups and Buildings
Title | Geometric and Topological Aspects of Coxeter Groups and Buildings PDF eBook |
Author | Anne Thomas |
Publisher | |
Pages | |
Release | 2018 |
Genre | |
ISBN | 9783037191897 |
The Geometry and Topology of Coxeter Groups. (LMS-32)
Title | The Geometry and Topology of Coxeter Groups. (LMS-32) PDF eBook |
Author | Michael Davis |
Publisher | Princeton University Press |
Pages | 600 |
Release | 2012-11-26 |
Genre | Mathematics |
ISBN | 1400845947 |
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.
Twin Buildings and Applications to S-Arithmetic Groups
Title | Twin Buildings and Applications to S-Arithmetic Groups PDF eBook |
Author | Peter Abramenko |
Publisher | Springer |
Pages | 131 |
Release | 2006-11-14 |
Genre | Mathematics |
ISBN | 3540495703 |
This book is addressed to mathematicians and advanced students interested in buildings, groups and their interplay. Its first part introduces - presupposing good knowledge of ordinary buildings - the theory of twin buildings, discusses its group-theoretic background (twin BN-pairs), investigates geometric aspects of twin buildings and applies them to determine finiteness properties of certain S-arithmetic groups. This application depends on topological properties of some subcomplexes of spherical buildings. The background of this problem, some examples and the complete solution for all "sufficiently large" classical buildings are covered in detail in the second part of the book.
Combinatorics of Coxeter Groups
Title | Combinatorics of Coxeter Groups PDF eBook |
Author | Anders Bjorner |
Publisher | Springer Science & Business Media |
Pages | 371 |
Release | 2006-02-25 |
Genre | Mathematics |
ISBN | 3540275967 |
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
Buildings
Title | Buildings PDF eBook |
Author | Kenneth S. Brown |
Publisher | Springer Science & Business Media |
Pages | 221 |
Release | 2013-06-29 |
Genre | Mathematics |
ISBN | 1461210194 |
For years I have heard about buildings and their applications to group theory. I finally decided to try to learn something about the subject by teaching a graduate course on it at Cornell University in Spring 1987. This book is based on the not es from that course. The course started from scratch and proceeded at a leisurely pace. The book therefore does not get very far. Indeed, the definition of the term "building" doesn't even appear until Chapter IV. My hope, however, is that the book gets far enough to enable the reader to tadle the literat ure on buildings, some of which can seem very forbidding. Most of the results in this book are due to J. Tits, who originated the the ory of buildings. The main exceptions are Chapter I (which presents some classical material), Chapter VI (which prcsents joint work of F. Bruhat and Tits), and Chapter VII (which surveys some applications, due to var ious people). It has been a pleasure studying Tits's work; I only hope my exposition does it justice.