Congruence Subgroups of Lattices in Rank 2 Kac–moody Groups over Finite Fields

Let G be a rank 2 complete affine or hyperbolic Kac–Moody group over a finite field k. Then G is locally compact and totally disconnected. Let B− = HU− be the negative minimal parabolic subgroup of G, where H is the analog of a diagonal subgroup and U− is generated by all negative real root groups. Let w1 and w2 be the generators of the Weyl group. Let P − 1 = B − t Bw1B be the negative standard parabolic subgroup of G corresponding to w1. It is known that the subgroups U −, B− and P− 1 are nonuniform lattice subgroups of G. Here we construct an infinite sequence of congruence subgroups of P− 1 as natural generalizations of the corresponding notions for lattices in Lie groups. We also show that certain subgroups of U− contain analogous congruence subgroups. Our technique involves determining graphs of groups presentations for U−, B− and P− 1 with the fundamental apartment of the Bruhat-Tits tree X a quotient graph for U− and for B− on X. When k = Fq and q = 2, the graph of groups for P− 1 has the the positive half of the fundamental apartment as quotient graph. We explicitly construct the graphs of groups for the principal (level 1) congruence subgroup of P− 1 and the analogous subgroups of U− giving generalized amalgam presentations for them. It follows that our congruence subgroups are nonuniform lattice subgroups of G. Dedicated to the memory of Eisa Abid whose short life touched many people.