Finiteness Properties of Arithmetic Groups Acting on Twin Buildings
Title | Finiteness Properties of Arithmetic Groups Acting on Twin Buildings PDF eBook |
Author | Stefan Witzel |
Publisher | Springer |
Pages | 128 |
Release | 2014-07-16 |
Genre | Mathematics |
ISBN | 3319064770 |
Providing an accessible approach to a special case of the Rank Theorem, the present text considers the exact finiteness properties of S-arithmetic subgroups of split reductive groups in positive characteristic when S contains only two places. While the proof of the general Rank Theorem uses an involved reduction theory due to Harder, by imposing the restrictions that the group is split and that S has only two places, one can instead make use of the theory of twin buildings.
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.
Groups
Title | Groups PDF eBook |
Author | Thomas Wolfgang Müller |
Publisher | Cambridge University Press |
Pages | 608 |
Release | 2004-04-08 |
Genre | Mathematics |
ISBN | 9780521542876 |
Survey and research articles from the Bielefeld conference on topological, combinatorial and arithmetic aspects of groups.
Tits Buildings and the Model Theory of Groups
Title | Tits Buildings and the Model Theory of Groups PDF eBook |
Author | Katrin Tent |
Publisher | Cambridge University Press |
Pages | 314 |
Release | 2002-01-03 |
Genre | Mathematics |
ISBN | 9780521010634 |
Introduction to buildings and their geometries with emphasis on model theoretic constructions, covering recent developments.
Algebraic Groups and Number Theory
Title | Algebraic Groups and Number Theory PDF eBook |
Author | Vladimir Platonov |
Publisher | Cambridge University Press |
Pages | 379 |
Release | 2023-08-31 |
Genre | Mathematics |
ISBN | 052111361X |
The first volume of a two-volume book offering a comprehensive account of the arithmetic theory of algebraic groups.
Algebraic Groups and Number Theory: Volume 1
Title | Algebraic Groups and Number Theory: Volume 1 PDF eBook |
Author | Vladimir Platonov |
Publisher | Cambridge University Press |
Pages | 380 |
Release | 2023-08-31 |
Genre | Mathematics |
ISBN | 1009380656 |
The first edition of this book provided the first systematic exposition of the arithmetic theory of algebraic groups. This revised second edition, now published in two volumes, retains the same goals, while incorporating corrections and improvements, as well as new material covering more recent developments. Volume I begins with chapters covering background material on number theory, algebraic groups, and cohomology (both abelian and non-abelian), and then turns to algebraic groups over locally compact fields. The remaining two chapters provide a detailed treatment of arithmetic subgroups and reduction theory in both the real and adelic settings. Volume I includes new material on groups with bounded generation and abstract arithmetic groups. With minimal prerequisites and complete proofs given whenever possible, this book is suitable for self-study for graduate students wishing to learn the subject as well as a reference for researchers in number theory, algebraic geometry, and related areas.
Geometry, Rigidity, and Group Actions
Title | Geometry, Rigidity, and Group Actions PDF eBook |
Author | Benson Farb |
Publisher | University of Chicago Press |
Pages | 659 |
Release | 2011-04-15 |
Genre | Mathematics |
ISBN | 0226237907 |
The study of group actions is more than a hundred years old but remains to this day a vibrant and widely studied topic in a variety of mathematic fields. A central development in the last fifty years is the phenomenon of rigidity, whereby one can classify actions of certain groups, such as lattices in semi-simple Lie groups. This provides a way to classify all possible symmetries of important spaces and all spaces admitting given symmetries. Paradigmatic results can be found in the seminal work of George Mostow, Gergory Margulis, and Robert J. Zimmer, among others. The papers in Geometry, Rigidity, and Group Actions explore the role of group actions and rigidity in several areas of mathematics, including ergodic theory, dynamics, geometry, topology, and the algebraic properties of representation varieties. In some cases, the dynamics of the possible group actions are the principal focus of inquiry. In other cases, the dynamics of group actions are a tool for proving theorems about algebra, geometry, or topology. This volume contains surveys of some of the main directions in the field, as well as research articles on topics of current interest.