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© 2002

Moufang Polygons

Benefits

  • Gives a complete classification of Moufang polygons, starting from first principles

  • Includes a totally new classification of the spherical buildings of rank 3 at least

  • J. Tits is one of the best and most influential algebraists of the past 50 years

  • R. Weiss is one of the world's leading researchers in the field of combinatorial group theory

  • The book will become a classic in the field

Book

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-x
  2. Preliminary Results

    1. Front Matter
      Pages 1-1
    2. Jacques Tits, Richard M. Weiss
      Pages 3-5
    3. Jacques Tits, Richard M. Weiss
      Pages 7-14
    4. Jacques Tits, Richard M. Weiss
      Pages 15-17
    5. Jacques Tits, Richard M. Weiss
      Pages 19-22
    6. Jacques Tits, Richard M. Weiss
      Pages 23-25
    7. Jacques Tits, Richard M. Weiss
      Pages 27-30
    8. Jacques Tits, Richard M. Weiss
      Pages 31-32
    9. Jacques Tits, Richard M. Weiss
      Pages 33-42
  3. Nine Families of Moufang Polygons

    1. Front Matter
      Pages 43-43
    2. Jacques Tits, Richard M. Weiss
      Pages 45-55
    3. Jacques Tits, Richard M. Weiss
      Pages 57-60
    4. Jacques Tits, Richard M. Weiss
      Pages 61-70
    5. Jacques Tits, Richard M. Weiss
      Pages 71-90
    6. Jacques Tits, Richard M. Weiss
      Pages 91-123
    7. Jacques Tits, Richard M. Weiss
      Pages 125-132
    8. Jacques Tits, Richard M. Weiss
      Pages 133-162
    9. Jacques Tits, Richard M. Weiss
      Pages 163-174
    10. Jacques Tits, Richard M. Weiss
      Pages 175-176

About this book

Introduction

Spherical buildings are certain combinatorial simplicial complexes intro­ duced, at first in the language of "incidence geometries," to provide a sys­ tematic geometric interpretation of the exceptional complex Lie groups. (The definition of a building in terms of chamber systems and definitions of the various related notions used in this introduction such as "thick," "residue," "rank," "spherical," etc. are given in Chapter 39. ) Via the notion of a BN-pair, the theory turned out to apply to simple algebraic groups over an arbitrary field. More precisely, to any absolutely simple algebraic group of positive rela­ tive rank £ is associated a thick irreducible spherical building of the same rank (these are the algebraic spherical buildings) and the main result of Buildings of Spherical Type and Finite BN-Pairs [101] is that the converse, for £ ::::: 3, is almost true: (1. 1) Theorem. Every thick irreducible spherical building of rank at least three is classical, algebraic' or mixed. Classical buildings are those defined in terms of the geometry of a classical group (e. g. unitary, orthogonal, etc. of finite Witt index or linear of finite dimension) over an arbitrary field or skew-field. (These are not algebraic if, for instance, the skew-field is of infinite dimension over its center. ) Mixed buildings are more exotic; they are related to groups which are in some sense algebraic groups defined over a pair of fields k and K of characteristic p, where KP eke K and p is two or (in one case) three.

Keywords

Buildings Graph algebra classification combinatorics generalized polygons graph theory incidence geometry polygon

Authors and affiliations

  1. 1.Collège de FranceParis Cedex 05France
  2. 2.Department of MathematicsTufts UniversityMedfordUSA

Bibliographic information

Reviews

From the reviews:

"In this excellently written book, the authors give a full classification for Moufang polygons. … The book is self contained … . the content of the book is accessible for motivated graduate students and researchers from every branch of mathematics. We recommend this book for everybody who is interested in the developments of the modern algebra, geometry or combinatorics." (Gábor P. Nagy, Acta Scientiarum Mathematicarum, Vol. 71, 2005)

"The publication of this long-awaited book is a major event for geometry in general, and for the theory of buildings in particular. … The classifications established in this book are splendid achievements of fundamental significance. The whole book is extremely well written, in a clear and concise style … . It is the definitive treatment and a standard reference." (Theo Grundhöfer, Mathematical Reviews, Issue 2003 m)

"This book contains the complete classification of all Moufang generalized polygons, including the full proof. … So, in conclusion, the book is a Bible for everyone interested in classification results related to spherical buildings. It is written in a very clear and concise way. It should be in the library of every mathematician as one of the major results in the theory of (Tits) buildings, (combinatorial) incidence geometry and (algebraic) group theory." (Hendrik Van Maldeghem, Bulletin of the Belgian Mathematical Society, Vol. 11 (3), 2005)