Cocompact lattices in complete Kac-Moody groups with Weyl group right-angled or a free product of spherical special subgroups

Abstract

Let G be a complete Kac-Moody group of rank n \geq 2 over the finite field of order q, with Weyl group W and building Δ. We first show that if W is right-angled, then for all q \neq 1 mod 4 the group G admits a cocompact lattice Γwhich acts transitively on the chambers of Δ. We also obtain a cocompact lattice for q =1 mod 4 in the case that Δis Bourdon's building. As a corollary of our constructions, for certain right-angled W and certain q, the lattice Γhas a surface subgroup. We also show that if W is a free product of spherical special subgroups, then for all q, the group G admits a cocompact lattice Γwith Γa finitely generated free group. Our proofs use generalisations of our results in rank 2 concerning the action of certain finite subgroups of G on Δ, together with covering theory for complexes of groups.

19 pages. Version 2: we have generalised from Weyl group a free product of cyclic groups of order 2 to the two cases indicated by the new title