Points and Lines Characterizing the Classical Geometries
The classical geometries of points and lines include not only the projective and polar spaces, but similar truncations of geometries naturally arising from the groups of Lie type. Virtually all of these geometries (or homomorphic images of them) are chara
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Ernest E. Shult
Points and Lines Characterizing the Classical Geometries
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Ernest E. Shult Kansas State University Denison Avenue 419 Manhattan KS 66502 USA [email protected]
Editorial board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Università degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus J. MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, École Polytechnique Endre Süli, University of Oxford Wojbor A. Woyczynski, Case Western Reserve University
ISBN 978-3-642-15626-7 e-ISBN 978-3-642-15627-4 DOI 10.1007/978-3-642-15627-4 Springer Heidelberg Dordrecht London New York Mathematics Subject Classification (2010): 51A45, 51A50, 51A05 c Springer-Verlag Berlin Heidelberg 2011 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. 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. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Professor Otto Kegel Professor Jacques Tits The von Humboldt Foundation
Preface
This book is about characterizing the classical geometries of Lie type by simple axioms on point and lines. If this book were written for experts only, it would be half as long as it is. Instead it is a teaching book. I have tried to write this book so that anyone – knowing very little of Incidence Geometry – could actually learn all about it. Ideally, without losing any connecting rapport, the graduate student beginner could be led on a path beginning at an elementary level involving familiar objects (graphs and groups, for example) right up to the research level. Of course that requires three things: 1. A vast amount of definitions, and lengthy explanations of where theorems are leading and why one should be led by them. 2. A presentation which, at each stage, seeks the most pedagogically efficient path for one whose only background is that obtained from preceeding stages. As a result, the route taken is somewhat different from standard treatments. (For example in the presentation of Tierlinck’s Theory, or in the definition of “building.”) 3. A self-contained account. As a result, the student will find almost everything
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