Covolumes of Uniform Lattices Acting on Polygonal Complexes

Let X be a polygonal complex with finitely many isometry classes of links. We establish a restriction on the covolumes of uniform lattices acting on X. When all links of X are isometric, and are either a complete bipartite graph or the building for a Chevalley group of rank 2 over a field of prime order, we obtain further restrictions on covolumes. Some of these restrictions are sharp: in [Th] we show that, for certain hyperbolic buildings, every rational number which satisfies these restrictions is the covolume of some uniform lattice.