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

Inna Capdeboscq and Anne Thomas

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 \(\Delta\). We first show that if \(W\) is right-angled, then for all \(q \not\equiv 1 \pmod 4\) the group \(G\)admits a cocompact lattice \(\Gamma\) which acts transitively on the chambers of \(\Delta\). We also obtain a cocompact lattice for \(q \equiv 1 \pmod 4\) in the case that \(\Delta\) is Bourdon's building. As a corollary of our constructions, for certain right-angled \(W\) and certain \(q\), the lattice \(\Gamma\) has 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 \(\Gamma\) with \(\Gamma\) a finitely generated free group. Our proofs use generalisations of our results in rank 2 concerning the action of certain finite subgroups of \(G\) on \(\Delta\), together with covering theory for complexes of groups.