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Lectures on Buildings

Updated and Revised

With a new Introduction by Mark Ronan

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

248 pages | 4 line drawings | 6 x 9 | © 2009

Mathematics and Statistics

Table of Contents

introduction to the 2009 edition



chapter 1 - chamber systems and examples

1. chamber systems

2. two examples of buildings


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


chapter 3 - buildings

1. a definition of buildings

2. generalised m-gons - the rank 2 case

3. residues and apartments


chapter 4 - local properties and coverings

1. chamber systems of type m

2. coverings and the fundamental group

3. the universal cover

4. examples


chapter 5 - bn - pairs

1. tits systems and buildings

2. parabolic subgroups


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


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


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)


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


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


index of notation


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