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

Abstract

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 be the negative standard parabolic subgroup of G corresponding to w₁. It is known that the subgroups U⁻, B⁻ and , are nonuniform lattice subgroups of G. Here we construct an infinite sequence of congruence subgroups of 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 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 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 and the analogous subgroups of U⁻ giving generalized amalgam presentations for them.

Key Words:

• Congruence subgroup
• Kac–Moody group
• Lattice

2010 Mathematics Subject Classification:

• 20E42
• 22E40
• 22F50

ACKNOWLEDGEMENT

The authors would like to thank Mikhail Ershov and Howard Garland for helpful discussions. We are also grateful to the referee whose comments improved the exposition.

Dedicated to the memory of Eisa Abid whose short life touched many people.