Hyperbolic Kac–Moody Weyl groups, billiard tables and actions of lattices on buildings

Let G be a locally compact Kac–Moody group associated to a symmetrizable rank 3 Kac–Moody algebra of noncompact hyperbolic type. It is known that the fundamental chambers of Weyl groups of certain algebras in this class serve as billiard tables for a physical phenomenon known as cosmological billiards. We discuss the classification of Weyl groups in this class; we determine their group presentations and exhibit their tessellations on the Poincaré disk. We show that each Weyl group W is an amalgam of finite Coxeter groups by constructing an action of W on a tree Y. It follows that among all Kac–Moody Weyl groups of affine or hyperbolic type, those of rank 2 and of rank 3 noncompact hyperbolic type are the only ones that have nontrivial amalgam decompositions. We show that the degrees of Y can be determined from the Dynkin diagram of G. We construct a bihomogeneous bipartite tree X embedded in the Tits building of G, a rank 3 locally finite hyperbolic building, on which the full Kac–Moody group G acts and we determine the degrees of X. We show that there is a nonuniform lattice subgroup of G which acts properly on this tree, that is, with finite vertex stabilizers.