PreprintCocompact lattices in complete KacMoody groups with Weyl group rightangled or a free product of spherical special subgroupsInna Capdeboscq and Anne ThomasAbstractLet \(G\) be a complete KacMoody 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 rightangled, 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 rightangled \(W\) 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. This paper is available as a pdf (412kB) file.
