## 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.

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 Tuesday, October 9, 2012