The paper under review studies lattices \(\Gamma\subset G_1\times\cdots\times G_n\) in products of (closed, co-compactly acting subgroups of) isometry groups of CAT(0)-spaces.
The main result states that if \(n\geq 3\), all \(G_i\) are irreducible complete Kac-Moody groups of simply laced type over a finite field and the projection of \(\Gamma\) to each \(G_i\) is faithful, then all \(G_i\) are topologically commensurable to semi-simple algebraic groups \(G_i'\) over a local field and \(\Gamma\) is an S-arithmetic lattice. (“Topologically commensurable” means that \(G_i\) contains a co-compact normal subgroup which is a compact extension of \(G_i'\).)
The proof relies on the following arithmeticity vs.non-residual-finiteness alternative. If \(n\geq 2\) and at least one factor is an infinite irreducible complete Kac-Moody group of simply laced type over a finite field, then either \(\Gamma\) is S-arithmetic or \(\Gamma\) is not residually finite.
As a consequence, topologically irreducible lattices in products of certain Kac-Moody groups (over sufficiently large finite fields) have discrete commensurators and are thus contained in a unique maximal lattice, and they satisfy an arithmeticity vs.simplicity alternative. More generally, the authors obtain that a finitely generated group \(\Gamma\) without non-trivial infinite index normal subgroup, which acts by isometries faithfully, minimally and without fixed point at infinity on an irreducible proper CAT(0)-space, must be either residually finite or virtually simple.
Section 2 also provides an extensive discussion about to which extent several notions of irreducibility for lattices \(\Gamma\subset G_1\times\cdots\times G_n\) are equivalent or not.