Lattices in Complete Rank 2 Kac–moody Groups
نویسنده
چکیده
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 [L] 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.
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