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 $n\ge 2$ over the finite field of order $q$, with Weyl group $W$ and building $\mathrm{\Delta }$. We first show that if $W$ is right-angled, then for all $q\not\equiv 1(\mathrm{mod}4)$ the group $G$admits a cocompact lattice $\mathrm{\Gamma }$ which acts transitively on the chambers of $\mathrm{\Delta }$. We also obtain a cocompact lattice for $q\equiv 1(\mathrm{mod}4)$ in the case that $\mathrm{\Delta }$ is Bourdon's building. As a corollary of our constructions, for certain right-angled $W$ and certain $q$, the lattice $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ with $\mathrm{\Gamma }$ 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 $\mathrm{\Delta }$, together with covering theory for complexes of groups.