CHAPTER 11

Buildings

Rudolf SCHARLAU

Fachbereich Mathematik, Universität Dortmund, D-44221 Dortmund, Germany

Contents

Introduction

1. Numbered complexes and chamber systems

Introduction

1.1. Numbered complexes: definitions and basic notions

1.2. Chamber systems

1.3. The relationship between numbered complexes and chamber systems

1.4. Numbered complexes and chamber systems with a group action

Notes to Section 1

2. Coxeter--Tits complexes

. Introduction

2.1. A survey of Coxeter groups

2.2. Thin complexes and reflections

2.3. Foldings and convexity

2.4. Characterizations of Coxeter--Tits complexes

2.5. Some calculations in Coxeter groups

Notes to Section 2

3. The axiomatics of buildings

. Introduction

3.1. Complexes belonging to a Coxeter diagram

3.2. Systems of apartments and buildings

3.3. The `First Main Characterization of Buildings'

3.4. Some applications of the `First Main Characterization'

Notes to Section 3

4. Some important classes of buildings

. Introduction

4.1. Buildings of type

4.2. Buildings of type and

4.3. Tits systems and buildings with a transitive group action

4.4. The building of a semisimple algebraic group

Notes to Section 4

5. On the geometry of buildings

. Introduction

5.1. Roots, convexity and projection maps

5.2. Opposite chambers and apartments in spherical buildings

5.3. Roots in spherical buildings and the Moufang property

Notes to Section 5

6. Buildings and covering theory

. Introduction

6.1. Coverings of chamber systems

6.2. The universal cover of a chamber system of Coxeter type

6.3. The `Second Main Characterization of Buildings'

6.4. Some applications, I: Geometrical axioms

6.5. Some applications, II: Group amalgamations

Notes to Section 6

7. The classification of buildings of irreducible spherical type and rank at least 3

. Introduction

7.1. Isomorphisms of spherical buildings

7.2. Constructing buildings from blueprints

7.3. The classification

Notes to Section 7

References