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

Let G be a rank 2 complete affine or hyperbolic simply-connected 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 (Formula presented.) be the negative standard parabolic subgroup of G corresponding to w₁. It is known that the subgroups U⁻, B⁻ and (Formula presented.) , are nonuniform lattice subgroups of G. Here we construct an infinite sequence of congruence subgroups of (Formula presented.) as natural generalizations of the corresponding notions for lattices in Lie groups. We also show that the group U⁻ contains analogous congruence subgroups. Our technique involves determining graphs of groups presentations for U⁻, B⁻, and (Formula presented.) with the fundamental apartment of the Bruhat–Tits tree X a quotient graph for U⁻ and for B⁻ on X. When k = 𝔽_q and q = 2^s, the graph of groups for (Formula presented.) 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 (Formula presented.) and the analogous subgroups of U⁻ giving generalized amalgam presentations for them.