From the preface: “This text started out as a revised version of Buildings by the second-named author… The current book includes all the material of the earlier one, but we have added a lot.” The outcome of the endeavour is an impressive seven-hundred-pages treatise on the theory of buildings.
The theory was created in the sixties, mainly by Jacques Tits, who was awarded the Abel prize in 2008 in part for this achievement. At the interface of algebra and geometry, the theory finds powerful applications to many areas, such as number theory, algebraic topology and representation theory. The size of Tits’ work is monumental (and often written in French), so it is really hard to learn the theory from the original papers, and one goes quickly in search of some self-contained account, like this book.
Here is a short summary of its contents. Three approaches to buildings are successively studied, so that the text can be separated in three parts, independent from each other to some extent.
The first approach is simplicial: one starts with a finite reflection group, generated by the reflections with respect to certain hyperplanes, which cut the space into polyhedral spaces. This is perhaps the most intuitive approach, especially in the case of finite Coxeter groups, which can be in fact realized in a canonical way as groups of orthogonal transformations in Euclidean spaces.
The second approach is combinatorial: one defines a building as an abstract set subject to certain axioms. Although this chapter is logically independent from the previous ones, an acquaintance with the results of the first approach is essential to understand the motivation of the list of axioms. This second approach, I think, is more amenable to the connection with graph theory, and is thus interesting in its own.
The third approach is metric: one defines chambers as certain metric spaces which can be glued together. It was M. W. Davis who discovered how to attach to every building a geometric realization that admits a CAT(0) metric (i.e., roughly speaking, a metric that makes it a Riemannian manifold with non-positive curvature).
Some of the (very important) applications of building theory to be found in this book are:
- Bruhat decompositions in classical groups,
- cohomological dimension of linear groups,
- discrete subgroups of p-adic groups,
- Mostow rigidity theorems.
For the applications to representation theory and harmonic analysis, so important in the Langlands program but beyond the scope of the book, the authors give a list of suggested readings.
Given the size of the book, it could be hardly studied in its entirety in a single course; however, one can certainly find several paths to build up a one-semester course on buildings, at an introductory or intermediate level. Necessary prerequisites include: group theory and linear algebra (of course), topology and some differential geometry. It is worth noting the presence of the three appendices, containing reminders on such topics as: Cell Complexes, Root Systems and Algebraic Groups. They will reassure when you feel lost.
Some of the numerous exercises have hints of solutions in the back of the book; there exists also a solutions manual, available from Springer’s Mathematical Editorial Department (but I have been too lazy to read it…).
As companions, I would suggest the following (wonderful) books:
- P. Garrett, Buildings and classical groups, Chapman & Hall
- J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press
- J-P. Serre, Trees, Springer-Verlag.
Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are mainly Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. At present, he works in a "classe préparatoire" in Geneva. He may be reached at firstname.lastname@example.org.