# Lectures on Buildings

## Updated and Revised

- Contents

introduction to the 2009 edition

introduction

leitfaden

chapter 1 - chamber systems and examples

1. chamber systems

2. two examples of buildings

exercises

chapter 2 - coxeter complexes

1. coxeter groups and complexes

2. words and galleries

3. reduced words and homotopy

4. finite coxeter complexes

5. self-homotopy

exercises

chapter 3 - buildings

1. a definition of buildings

2. generalised m-gons - the rank 2 case

3. residues and apartments

exercises

chapter 4 - local properties and coverings

1. chamber systems of type m

2. coverings and the fundamental group

3. the universal cover

4. examples

exercises

chapter 5 - bn - pairs

1. tits systems and buildings

2. parabolic subgroups

exercises

chapter 6 - buildings of spherical type and root groups

1. some basic lemmas

2. root groups and the moufang property

3. commutator relations

4. moufang buildings - the general case

exercises 80

chapter 7 - a construction of buildings

1. blueprints

2. natural labellings of moufang buildings

3. foundations

exercises

chapter 8 - the classification of spherical buildings

1.a3 blueprints and foundations

2. diagrams with single bonds

3. c3 foundations

4. cn buildings for n > 4

5. tits diagrams and f4 buildings

6. finite buildings

exercises

chapter 9 - affine buildings I

1. affine coxeter complexes and sectors

2. the affine building an-1 (k,v)

3. the spherical building at infinity

4. the proof of (9.5)

exercises

chapter 10 - affine buildings II

1. apartment systems, trees and projective valuations

2. trees associated to walls and panels at infinity

3. root groups with a valuation

4. construction of an affine bn-pair

5. the classification

6. an application

exercises

chapter 11 - twin buildings

1. twin buildings and kac-moody groups

2. twin trees

3. twin apartments

4. an example: affine twin buildings

5. residues, rigidity, and proj.

6. 2-spherical twin buildings

7. the moufang property and root group data

8. twin trees again

appendix 1 - moufang polygons

1. the m-function

2. the natural labelling for a moufang plane

3. the non-existence theorem

appendix 2 - diagrams for moufang polygons

appendix 3 - non-discrete buildings

appendix 4 - topology and the steinberg representation

appendix 5 - finite coxeter groups

appendix 6 finite buildings and groups of lie type

bibliography

index of notation

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