Lattices in Hyperbolic Buildings

This survey is intended as a brief introduction to the theory of hyperbolic buildings and their lattices. Hyperbolic buildings are negatively curved geometric objects which also have a rich algebraic and combinatorial structure, and the study of these buildings and the lattices in their automorphism groups involves a fascinating mixture of techniques from many different areas of mathematics. Roughly speaking, a hyperbolic building is obtained by gluing together many hyperbolic spaces which are tiled by polyhedra. For the precise definition, together with background on general buildings and known constructions of hyperbolic buildings, see Section 1 below. Given a hyperbolic building ∆, we write G = Aut(∆) for the group of automorphisms, or cellular isometries, of ∆. When the building ∆ is locally finite, the group G equipped with the compact-open topology is naturally a locally compact topological group, and so has a Haar measure μ. In this topology on G, a subgroup Γ < G is discrete if and only if it acts on ∆ with finite cell stabilisers. A lattice in G is a discrete subgroup Γ < G such that μ(Γ\G) <∞, and a lattice Γ is cocompact (or uniform) if Γ\G is compact. The Haar measure μ on G may be normalised so that the covolume μ(Γ\G) of a lattice Γ < G is given by the formula