Lattices in complete rank 2 Kac–Moody groups

Abstract

Let Λ be a minimal Kac–Moody group of rank 2 defined over the finite field Fq, where q=pa with p prime. Let G be the topological Kac–Moody group obtained by completing Λ. An example is G=SL2(K), where K is the field of formal Laurent series over Fq. The group G acts on its Bruhat–Tits building X, a tree, with quotient a single edge. We construct new examples of cocompact lattices in G, many of them edge-transitive. We then show that if cocompact lattices in G do not contain p-elements, the lattices we construct are the only edge-transitive lattices in G, and that our constructions include the cocompact lattice of minimal covolume in G. We also observe that, with an additional assumption on p-elements in G, the arguments of Lubotzky (1990) [21] for the case G=SL2(K) may be generalised to show that there is a positive lower bound on the covolumes of all lattices in G, and that this minimum is realised by a non-cocompact lattice, a maximal parabolic subgroup of Λ.