Buildings

Abstract

Let Δ be a complex, and let α be a set of subcoraplexes of Δ. The pair (Δ, α) is called a building of which the elements of α are called apartments if the following conditions hold:

• (Bl) Δ is thick;

• (B2) The elements of α are thin chamber complexes;

• (B3) Any two elements of Δ belong to an apartment;

• (B4) If two apartments Σ and Σ′ contain two elements A, A′ ∈ Δ, there exists an isomorphism of Σ onto Σ′ which leaves invariant A, A′ and all their faces.

It is clear that Δ is a chamber complex and that the apartments are isomorphic subcomplexes. We shall see (3.15) that the isomorphism class of the apartments is entirely determined by Δ. More precisely, it can be shown that if a complex Δ possesses a set α of subcomplexes such that (Δ, α) is a building, the union of all such sets α has the same property; in particular, a given complex Δ has at most one “maximal building structure”. In 3.26, we shall see that, if the diameter of Δ is finite, the set α is unique (when it exists). All this justifies the abuse of language which we shall often make by talking of “the building Δ”. Any representative of the isomorphism class of thin complexes to which belong the apartments of Δ will be called “the” Weyl complex of Δ.