Connectivity of chamber graphs of buildings and related complexes

Let ΔΔ 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)G(Δ), whose edges are the pairs of adjacent chambers in ΔΔ, is known to be qq-regular for a certain number q=q(Δ)q=q(Δ). Our main result is that G(Δ)G(Δ) is qq-connected in the sense of graph theory. In the language of building theory this means that every pair of chambers of ΔΔ is connected by qq pairwise disjoint galleries.Similar results are proved for the chamber graphs of Coxeter complexes and for order complexes of geometric lattices.