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 infinity, 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 congruent to Mq * (Mq boolean AND(M) over tildeq) (M) over tildeq with quotient a simplex, and we show that this condition is satisfied when q = 2(s). If further Mq and (M) over tildeq are abelian, we give a method for constructing an infinite descending chain of cocompact lattices ... Gamma 3 <= Gamma 2 <= Gamma 1 <= Gamma. This allows us to characterize each of the quotient graphs of groups Gamma(i)\\X, the presentations of the Gamma(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 Gamma'(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 Gamma'1 =. Lambda' for a non-discrete subgroup. Lambda' <= G whose quotient. Lambda' \X is equal to G\X. Using the action of Gamma'(1) on chi we construct an infinite descending chain of cocompact lattices ... Gamma'(3) <= Gamma'(2) <= Gamma'(1) in G. We also determine the quotient graphs of groups Gamma'(i)\\X, the presentations of the Gamma'(i) and their covolumes.