Mathematical Research Letters
Volume 20 (2013)
Co-compact lattices in complete Kac-Moody groups with Weyl group right-angled or a free product of spherical special subgroups
Pages: 339 – 358
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 $q \not \equiv 1$ (mod 4) the group $G$ admits a co-compact lattice $\Gamma$ which acts transitively on the chambers of $\Delta$. We also obtain a co-compact lattice for $q \equiv 1$ (mod 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. Our second main result states that if $W$ is a free product of spherical special subgroups, then for all $q$, the group $G$ admits a co-compact lattice $\Gamma$, with $\Gamma$ a finitely generated free group. Thus for many $G$ of rank $n \geq 3$, our results provide the first (explicit) constructions of cocompact lattices in $G$, and in particular, the first explicit examples of cocompact lattices in complete Kac-Moody groups whose Weyl group is not right-angled. Our proofs use generalizations 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.