Capdeboscq, Inna (Korchagina) and Thomas, Anne (2012) Lattices in complete rank 2 Kac–Moody groups. Journal of Pure and Applied Algebra, Vol.216 (No.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:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title:	Journal of Pure and Applied Algebra
Publisher:	Elsevier Science BV
ISSN:	00224049
Official Date:	June 2012
Dates:	Date Event June 2012 Published
Volume:	Vol.216
Number:	No.6
Page Range:	pp. 1348-1371
DOI:	10.1016/j.jpaa.2011.10.018
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access

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