Infinite Descending Chains of Cocompact Lattices in Kac-moody Groups

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 Γ ∼= Mq ∗Mq∩M̃q M̃q with quotient a simplex, and we show that this condition is satisfied when q = 2. If further Mq and M̃q 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 Γi 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 X . The tree X 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′1 on X 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 Γ′i\\X , the presentations of the Γ ′ i and their covolumes.