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• W2037181504 abstract "Let A be a symmetrizable affine or hyperbolic generalized Cartan matrix. Let G be a locally compact Kac–Moody group associated to A over a finite field 𝔽 q . We suppose that G has type ∞, that is, the Weyl group W of G is a free product of ℤ/2ℤ's. This includes all locally compact Kac–Moody groups of rank 2 and three possible locally compact rank 3 Kac–Moody groups of noncompact hyperbolic type. For every prime power q, we give a sufficient condition for the rank 2 Kac–Moody group G to contain a cocompact lattice [Formula: see text] with quotient a simplex, and we show that this condition is satisfied when q = 2 s . If further M q and [Formula: see text] are abelian, we give a method for constructing an infinite descending chain of cocompact lattices … Γ 3 ≤ Γ 2 ≤ Γ 1 ≤ Γ. This allows us to characterize each of the quotient graphs of groups Γ i X, the presentations of the Γ i and their covolumes, where X is the Tits building of G, a homogeneous tree. Our approach is to extend coverings of edge-indexed graphs to covering morphisms of graphs of groups with abelian groupings. This method is not specific to cocompact lattices in Kac–Moody groups and may be used to produce chains of subgroups acting on trees in a general setting. It follows that the lattices constructed in the rank 2 Kac–Moody group have the Haagerup property. When q = 2 and rank (G) = 3 we show that G contains a cocompact lattice Γ′ 1 that acts discretely and cocompactly on a simplicial tree [Formula: see text]. The tree [Formula: see text] is naturally embedded in the Tits building X of G, a rank 3 hyperbolic building. Moreover Γ′ 1 ≤ Λ′ for a non-discrete subgroup Λ′ ≤ G whose quotient Λ′ X is equal to GX. Using the action of Γ′ 1 on [Formula: see text] we construct an infinite descending chain of cocompact lattices …Γ′ 3 ≤ Γ′ 2 ≤ Γ′ 1 in G. We also determine the quotient graphs of groups [Formula: see text], the presentations of the Γ′ i and their covolumes." @default.
• W2037181504 created "2016-06-24" @default.
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• W2037181504 date "2011-12-01" @default.
• W2037181504 modified "2023-09-25" @default.
• W2037181504 title "INFINITE DESCENDING CHAINS OF COCOMPACT LATTICES IN KAC–MOODY GROUPS" @default.
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• W2037181504 doi "https://doi.org/10.1142/s0219498811005130" @default.
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