Twin Buildings and Applications to S-Arithmetic Groups
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
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Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Peter Abramenko
Twin Buildings and Applications to S-Arithmetic Groups
Springer
Author Peter Abramenko Fachbereich Mathematik Universitat Frankfurt Robert-Mayer-Str, 6-10 D-60054 Frankfurt, Germany e-mail: [email protected]
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme Abramenko, Peter: Twin buildings and applications to S-arithmetic groups / Peter Abramenko. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris ; Santa Clara; Singapore; Tokyo: Springer. 1996 (Lecture notes in mathematics ; 1641) ISBN 3-540-61973-9 NE:GT
Mathematics Subject Classification (1991): Primary: 20E42, 51E24, 20F32, 20030, 20105 Secondary: 05E25, 14L35, 51A50, IIE39, 51E12, 20F05, 20F55, 20G25, 22E67,17B67 ISSN 0075-8434 ISBN 3-540-61973-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1996 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10520133 46/3142-543210 - Printed on acid-free paper
To my mother and to my brother
Preface The present Lecture Notes volume combines aspects of two mathematical domains which are closely connected to each other: group theory and the theory of buildings. On the basis of investigations concerning twin buildings and sub complexes of spherical buildings, finiteness properties of some Sarithmetic groups are derived (d. the introduction for more details). Large parts of this book are devoted to the theory of (twin) buildings and not only written with group theoretic applications in mind. The first two sections of Chapter I can serve very well as an introduction to twin buildings. §1 describes the group theoretic background of this new theory. The basic definitions and facts (see in particular Lemma 2) are introduced in §2. Though these results are mainly due to Tits, the complete proofs are given here since they are hard to find in the original papers or not yet published. The following two sections present some of my own investigations concerning twin buildings. These are applied at the e
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