Lattices in Trees and Higher Dimensional Complexes

Let G be a locally compact topological group. A lattice in G is a discrete subgroup Γ such that Γ\G carries a finite G–invariant measure, and Γ is uniform or cocompact if Γ\G is compact. Lattices in Lie groups have been well-studied. See, for example, Raghunathan [48], and for open problems the section on “Lattices in Lie groups” in this wiki. Much less is known about lattices in other locally compact groups. We consider lattices in the following setting. Let X be a locally finite polyhedral complex, such as a tree, a product of trees, or a (classical or nonclassical) building. Let G = Aut(X) be the group of automorphisms, or cellular isometries, of X. With the compact-open topology, G is naturally a locally compact group. Provided X\G is finite, a discrete subgroup Γ ≤ G is a lattice if and only if the series ∑ x∈Γ\V X |Γx|−1 converges (Serre [8]). Much work on lattices in Aut(X) has been motivated by finding similarities and differences with lattices in Lie groups. Methods of geometric group theory have so far proved useful.