AbstractsMathematics

We utilize graphs of groups and the corresponding covering theory to study lattices in type-infinity Kac-Moody groups over a finite field of size q, including results for both cocompact and nonuniform lattices. For every prime power q we give a sufficient condition for the rank 2 Kac-Moody group G to contain a cocompact lattice with quotient a simplex, and we show that this condition is satisfied when q is a power of 2. Under further restrictions, we show that there is an infinite descending chain of cocompact lattices, and we demonstrate such a chain for q=2. Moreover we characterize the quotient graphs of groups for each lattice. Our method involves extending coverings of edge-indexed graphs to covering morphisms of graphs of groups. We also show how this gives rise to other infinite families of cocompact lattices in G. When q=2 we are also able to embed the infinite descending chain in the rank 3 Kac-Moody group as a chain of lattices in the subgroup generated by all non-maximal standard parabolic subgroups. In addition we embed a non-discrete subgroup in the rank 3 Kac-Moody group whose quotient is a simplex. We next give graphs of groups descriptions for known nonuniform lattices of Nagao-type. For the nonuniform lattices SL_2 and PGL_2 over polynomial rings with base field F_q we use the theory of ramified coverings to construct the graphs of groups for their congruence subgroups. We also examine the same construction employed by Morgenstern, identifying and repairing an error in his work. All graphs of groups for non-uniform lattices constructed here satisfy the structure theorem for graphs of groups.