# Lectures on Buildings

## Updated and Revised

In mathematics, “buildings” are geometric structures that represent groups of Lie type over an arbitrary field. This concept is critical to physicists and mathematicians working in discrete mathematics, simple groups, and algebraic group theory, to name just a few areas.

Almost twenty years after its original publication, Mark Ronan’s *Lectures on Buildings *remains one of the best introductory texts on the subject. A thorough, concise introduction to mathematical buildings, it contains problem sets and an excellent bibliography that will prove invaluable to students new to the field. *Lectures on Buildings *will find a grateful audience among those doing research or teaching courses on Lie-type groups, on finite groups, or on discrete groups.

“Ronan’s account of the classification of affine buildings [is] both interesting and stimulating, and his book is highly recommended to those who already have some knowledge and enthusiasm for the theory of buildings.”—*Bulletin of the London Mathematical Society*

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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