Download or read book Buildings and Classical Groups written by Paul B. Garrett and published by CRC Press. This book was released on 1997-04-01 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: Buildings are highly structured, geometric objects, primarily used in the finer study of the groups that act upon them. In Buildings and Classical Groups, the author develops the basic theory of buildings and BN-pairs, with a focus on the results needed to apply it to the representation theory of p-adic groups. In particular, he addresses spherical and affine buildings, and the "spherical building at infinity" attached to an affine building. He also covers in detail many otherwise apocryphal results. Classical matrix groups play a prominent role in this study, not only as vehicles to illustrate general results but as primary objects of interest. The author introduces and completely develops terminology and results relevant to classical groups. He also emphasizes the importance of the reflection, or Coxeter groups and develops from scratch everything about reflection groups needed for this study of buildings. In addressing the more elementary spherical constructions, the background pertaining to classical groups includes basic results about quadratic forms, alternating forms, and hermitian forms on vector spaces, plus a description of parabolic subgroups as stabilizers of flags of subspaces. The text then moves on to a detailed study of the subtler, less commonly treated affine case, where the background concerns p-adic numbers, more general discrete valuation rings, and lattices in vector spaces over ultrametric fields. Buildings and Classical Groups provides essential background material for specialists in several fields, particularly mathematicians interested in automorphic forms, representation theory, p-adic groups, number theory, algebraic groups, and Lie theory. No other available source provides such a complete and detailed treatment.

Download or read book Euclidean Buildings written by Guy Rousseau and published by European Mathematical Society. This book was released on 2023-08-15 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of buildings lies at the interplay between geometry and group theory and is one of the main tools for studying the structure of many groups. Actually, buildings were introduced by Jacques Tits in the 1950s to better understand and study a semi-simple algebraic group over a field. For a general field, its associated building is a spherical building, called its Tits building. It is a simplicial complex and, in this book, one considers a geometric realization called vectorial building. When the field is real valued, François Bruhat and Jacques Tits constructed another building taking into account the topology of the field. This Bruhat-Tits building is a polysimplicial complex and its usual geometric realization is an affine building. These vectorial or affine buildings are the main examples of Euclidean buildings. This book develops the general abstract theory of these Euclidean buildings (the buildings with Euclidean affine spaces as apartments). It is largely self contained and emphasizes the metric aspects of these objects, as CAT(0) spaces very similar to Riemannian symmetric spaces of non-compact type. The book studies their compactifications, their links with groups, many classical examples, and some applications (for example, to Hecke algebras).

Download or read book Descent in Buildings AM 190 written by Bernhard Mühlherr and published by Princeton University Press. This book was released on 2015-09-22 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: Descent in Buildings begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. The authors then put their algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or "form" of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a "residually pseudo-split" Bruhat-Tits building. The book concludes with a display of the Tits indices associated with each of these exceptional forms. This is the third and final volume of a trilogy that began with Richard Weiss' The Structure of Spherical Buildings and The Structure of Affine Buildings.

Download or read book Lectures on Buildings written by Mark Ronan and published by University of Chicago Press. This book was released on 2009-10-15 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: In mathematics, “buildings” are geometric structures that represent groups of Lie type over an arbitrary field. This concept is critical to physicists and mathematicians working in discrete mathematics, simple groups, and algebraic group theory, to name just a few areas. Almost twenty years after its original publication, Mark Ronan’s Lectures on Buildings remains one of the best introductory texts on the subject. A thorough, concise introduction to mathematical buildings, it contains problem sets and an excellent bibliography that will prove invaluable to students new to the field. Lectures on Buildings will find a grateful audience among those doing research or teaching courses on Lie-type groups, on finite groups, or on discrete groups. “Ronan’s account of the classification of affine buildings [is] both interesting and stimulating, and his book is highly recommended to those who already have some knowledge and enthusiasm for the theory of buildings.”—Bulletin of the London Mathematical Society

Download or read book Buildings written by Kenneth S. Brown and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Twin Buildings and Applications to S Arithmetic Groups written by Peter Abramenko and published by Springer. This book was released on 2006-11-14 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Compactifications of Symmetric Spaces written by Yves Guivarc'h and published by Springer Science & Business Media. This book was released on 1998-08-26 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: The concept of symmetric space is of central importance in many branches of mathematics. Compactifications of these spaces have been studied from the points of view of representation theory, geometry, and random walks. This work is devoted to the study of the interrelationships among these various compactifications and, in particular, focuses on the martin compactifications. It is the first exposition to treat compactifications of symmetric spaces systematically and to uniformized the various points of view. The work is largely self-contained, with comprehensive references to the literature. It is an excellent resource for both researchers and graduate students.

Download or read book Tits Buildings and the Model Theory of Groups written by Katrin Tent and published by Cambridge University Press. This book was released on 2002-01-03 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction to buildings and their geometries with emphasis on model theoretic constructions, covering recent developments.

Download or read book The Structure of Affine Buildings AM 168 written by Richard Mark Weiss and published by Princeton University Press. This book was released on 2009 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: Richard Weiss gives a detailed presentation of the complete proof of the classification of Bruhat-Tits buildings first completed by Jacques Tits in 1986. The book includes numerous results about automorphisms, completions and residues of these buildings.

Download or read book Buildings and Classical Groups written by Paul B. Garrett and published by Springer. This book was released on 2012-11-06 with total page 373 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes the structure of the classical groups, meaning general linear groups, symplectic groups, and orthogonal groups, both over general fields and in finer detail over p-adic fields. To this end, half of the text is a systematic development of the theory of buildings and BN-pairs, both spherical and affine, while the other half is illustration by and application to the classical groups. The viewpoint is that buildings are the fundamental objects, used to study groups which act upon them. Thus, to study a group, one discovers or con structs a building naturally associated to it, on which the group acts nicely. This discussion is intended to be intelligible after completion of a basic graduate course in algebra, so there are accounts of the necessary facts about geometric algebra, reflection groups, p-adic numbers (and other discrete val uation rings), and simplicial complexes and their geometric realizations. It is worth noting that it is the building-theoretic aspect, not the algebraic group aspect, which determines the nature of the basic representation theory of p-adic reductive groups.

Download or read book Buildings Finite Geometries and Groups written by N.S. Narasimha Sastry and published by Springer Science & Business Media. This book was released on 2011-11-13 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the Proceedings of the ICM 2010 Satellite Conference on “Buildings, Finite Geometries and Groups” organized at the Indian Statistical Institute, Bangalore, during August 29 – 31, 2010. This is a collection of articles by some of the currently very active research workers in several areas related to finite simple groups, Chevalley groups and their generalizations: theory of buildings, finite incidence geometries, modular representations, Lie theory, etc. These articles reflect the current major trends in research in the geometric and combinatorial aspects of the study of these groups. The unique perspective the authors bring in their articles on the current developments and the major problems in their area is expected to be very useful to research mathematicians, graduate students and potential new entrants to these areas.

Download or read book Group Theory From A Geometrical Viewpoint written by Alberto Verjovski and published by #N/A. This book was released on 1991-08-12 with total page 744 pages. Available in PDF, EPUB and Kindle. Book excerpt: This proceedings presents the latest research materials done on group theory from geometrical viewpoint in particular Gromov's theory of hyperbolic groups, Coxeter groups, Tits buildings and actions on real trees. All these are very active subjects.

Download or read book The Structure of Spherical Buildings written by Richard M. Weiss and published by Princeton University Press. This book was released on 2020-07-21 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a clear and authoritative introduction to the theory of buildings, a topic of central importance to mathematicians interested in the geometric aspects of group theory. Its detailed presentation makes it suitable for graduate students as well as specialists. Richard Weiss begins with an introduction to Coxeter groups and goes on to present basic properties of arbitrary buildings before specializing to the spherical case. Buildings are described throughout in the language of graph theory. The Structure of Spherical Buildings includes a reworking of the proof of Jacques Tits's Theorem 4.1.2. upon which Tits's classification of thick irreducible spherical buildings of rank at least three is based. In fact, this is the first book to include a proof of this famous result since its original publication. Theorem 4.1.2 is followed by a systematic study of the structure of spherical buildings and their automorphism groups based on the Moufang property. Moufang buildings of rank two were recently classified by Tits and Weiss. The last chapter provides an overview of the classification of spherical buildings, one that reflects these and other important developments.

Download or read book Collected works written by Jacques Tits and published by Ashgate Publishing, Ltd.. This book was released on 2013-11-15 with total page 1032 pages. Available in PDF, EPUB and Kindle. Book excerpt: Jacques Tits was awarded the Wolf Prize in 1993 and the Abel Prize (jointly with John Thompson) in 2008. The impact of his contributions in algebra, group theory and geometry made over a span of more than five decades is incalculable. Many fundamental developments in several fields of mathematics have their origin in ideas of Tits. A number of Tits' papers mark the starting point of completely new directions of research. Outstanding examples are papers on quadratic forms, on Kac-Moody groups and on what subsequently became known as the Tits alternative. These volumes contain an almost complete collection of Tits' mathematical writings. They include, in particular, a number of published and unpublished manuscripts which have not been easily accessible until now. This collection of Tits' contributions in one place makes the evolution of his mathematical thinking visible. The development of his theory of buildings and BN-pairs and its bearing on the theory of algebraic groups, for example, reveal a fascinating story. Along with Tits' mathematical writings, these volumes contain biographical data, survey articles on aspects of Tits' work, and comments by the editors on the content of some of his papers. With the publication of these volumes, a major piece of 20th-century mathematics is being made available to a wider audience.

Download or read book A Compactification of the Bruhat Tits Building written by Erasmus Landvogt and published by Springer. This book was released on 2006-11-14 with total page 159 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this work is the definition of the polyhedral compactification of the Bruhat-Tits building of a reductive group over a local field. In addition, an explicit description of the boundary is given. In order to make this work as self-contained as possible and also accessible to non-experts in Bruhat-Tits theory, the construction of the Bruhat-Tits building itself is given completely.

Download or read book A Compactification of the Bruhat Tits Building written by Erasmus Landvogt and published by Lecture Notes in Mathematics. This book was released on 1995-11-17 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this work is the definition of the polyhedral compactification of the Bruhat-Tits building of a reductive group over a local field. In addition, an explicit description of the boundary is given. In order to make this work as self-contained as possible and also accessible to non-experts in Bruhat-Tits theory, the construction of the Bruhat-Tits building itself is given completely.

Download or read book Finiteness Properties of Arithmetic Groups Acting on Twin Buildings written by Stefan Witzel and published by Springer. This book was released on 2014-07-16 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.