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Let G be a complete Kac-Moody group of rank n \geq 2 over the finite field oforder q, with Weyl group W and building \Delta. We first show that if W isright-angled, then for all q \neq 1 mod 4 the group G admits a cocompactlattice \Gamma which acts transitively on the chambers of \Delta. We alsoobtain a cocompact lattice for q =1 mod 4 in the case that \Delta is Bourdon'sbuilding. As a corollary of our constructions, for certain right-angled W andcertain q, the lattice \Gamma has a surface subgroup. We also show that if W isa free product of spherical special subgroups, then for all q, the group Gadmits a cocompact lattice \Gamma with \Gamma a finitely generated free group.Our proofs use generalisations of our results in rank 2 concerning the actionof certain finite subgroups of G on \Delta, together with covering theory forcomplexes of groups.

Cocompact lattices in complete Kac-Moody groups with Weyl group right-angled or a free product of spherical special subgroups