Covering theory of buildings and their quotients

Abstract

We introduce structures which model the quotients of Bruhat-Tits buildings by typepreserving group actions. These structures, which we call Weyl graphs, generalize chamber systems of type M by allowing 2-residues to be quotients of generalized polygons. Weyl graphs also generalize Tits amalgams with a trivial chamber stabilizer group by allowing for group actions which are not chamber-transitive. We develop covering theory of Weyl graphs, and characterize buildings as connected, simply connected Weyl graphs. We describe a procedure for obtaining a group presentation of the fundamental group of a Weyl graph W, which acts naturally on the universal cover of W. We present an application of the theory of Weyl graphs to Singer lattices. We construct the Singer cyclic lattices of type M, where mst 2 f2; 3;1g for all s; t 2 S. In particular, by taking the Davis realization of a building, we obtain new examples of lattices in polyhedral complexes.