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 픽q. 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 $\Gamma \cong M_q \ast_{M_q\cap \widetilde{M_q}}\widetilde{M_q}$ with quotient a simplex, and we show that this condition is satisfied when q = 2s. If further Mq and $\widetilde{M_q}$ 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 $\mathcal{X}$. The tree $\mathcal{X}$ 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 $\mathcal{X}$ 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 $\Gamma_i'\backslash\!\backslash\mathcal{X}$, the presentations of the Γ′i and their covolumes. [ABSTRACT FROM AUTHOR]