  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 \neq 1 mod 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 =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. 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.
