Connectivity of chamber graphs of buildings and related complexes

Let \Delta be a finite building (or, more generally, a thick spherical and locally finite building). The chamber graph G(\Delta), whose edges are the pairs of adjacent chambers in \Delta, is known to be q-regular for a certain number q=q(\Delta). Our main result is that G(\Delta) is q-connected in the sense of graph theory. Similar results are proved for the chamber graphs of Coxeter complexes and for order complexes of geometric lattices.