12 - The Classification of affine buildings

Summary

Introduction

The theory of affine buildings reveals fascinating links between group theory, Euclidean geometry and number theory. In particular, reflections, the Weyl chambers of root systems and valuations of fields all play a central role in their classification.

The study of affine buildings was begun by Bruhat and Tits in [4] and the classification of affine buildings of rank at least four was completed by Tits in [11]. When combined with the classification of Moufang polygons carried out in [12], the Bruhat-Tits classification covers also affine buildings of rank three under the assumption in this case that the building at infinity is Moufang.

Our goal here is to give a very brief overview of this work. All the details can be found in the forthcoming book [14] (as well, of course, as in [4] and [11]).

In this article we regard buildings exclusively as certain edge-colored graphs. For different points of view, see [1]. Other excellent sources of results about affine buildings are [3], [5] and [8].

Buildings

Let Σ be an edge-colored graph and let I denote the set of colors appearing on the edges of Σ. We call |I| the rank of Σ. For each subset J of I let Σ_J be the graph obtained from Σ by deleting all the edges whose color is not in J (but without deleting any vertices). A J-residue of Σ for some subset J of I is a connected component of the graph Σ_J. Thus two distinct J-residues (for a fixed subset J of I) are always disjoint.