The Structure of Spherical Buildings

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.

Preface ix Chapter 1. Chamber Systems 1 Chapter 2. Coxeter Groups 9 Chapter 3. Roots 17 Chapter 4. Reduced Words 27 Chapter 5. Opposites 33 Chapter 6. 2-Interiors 41 Chapter 7. Buildings 47 Chapter 8. Apartments 61 Chapter 9. Spherical Buildings 73 Chapter 10. Extensions of Isometries 81 Chapter 11. The Moufang Property 91 Chapter 12. Root Group Labelings 117 References 131 Index 133