Connectivity of chamber graphs of buildings and related complexes

Let Δ be a thick and locally finite building with the property that no edge of the associated Coxeter diagram has label “ ∞ ”. The chamber graph G ( Δ ) , whose edges are the pairs of adjacent chambers in Δ , is known to be q -regular for a certain number q = q ( Δ ) . Our main result is that G ( Δ ) is q -connected in the sense of graph theory. In the language of building theory this means that every pair of chambers of Δ is connected by q pairwise disjoint galleries. r results are proved for the chamber graphs of Coxeter complexes and for order complexes of geometric lattices.