INFINITE DESCENDING CHAINS OF COCOMPACT LATTICES IN KAC–MOODY GROUPS

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 𝔽q. 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 = 2s. If further Mqand [Formula: see text] are abelian, we give a method for constructing an infinite descending chain of cocompact lattices … Γ3≤ Γ2≤ Γ1≤ Γ. This allows us to characterize each of the quotient graphs of groups Γi\\X, the presentations of the Γiand 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′1that 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 G\X. Using the action of Γ′1on [Formula: see text] we construct an infinite descending chain of cocompact lattices …Γ′3≤ Γ′2≤ Γ′1in G. We also determine the quotient graphs of groups [Formula: see text], the presentations of the Γ′iand their covolumes.