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

TL;DR

This paper constructs cocompact lattices in complete Kac-Moody groups with specific Weyl groups, demonstrating new lattice existence results and subgroup structures depending on the Weyl group and field size.

Contribution

It introduces new methods to construct cocompact lattices in Kac-Moody groups with right-angled or free product Weyl groups, expanding understanding of their subgroup structures.

Findings

Existence of cocompact lattices for right-angled Weyl groups when q ≠ 1 mod 4

Construction of lattices with surface subgroups for certain parameters

Finitely generated free lattices in cases with free product Weyl groups

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 \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.