Lattices in complete rank 2 Kac–Moody groups

Capdeboscq, I.K. and Thomas, A. (2012) Lattices in complete rank 2 Kac–Moody groups. Journal of Pure and Applied Algebra, 216(6), pp. 1348-1371. (doi: 10.1016/j.jpaa.2011.10.018)

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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 , 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 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.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Thomas, Dr Anne
Authors: Capdeboscq, I.K., and Thomas, A.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal of Pure and Applied Algebra
ISSN:0022-4049
ISSN (Online):1873-1376

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