Finite presentability of twisted Brin-Thompson groups

Given a group G acting faithfully on a set S, we characterize precisely when the twisted Brin-Thompson group SVG is finitely presented. The answer is that SVG is finitely presented if and only if we have the following: G is finitely presented, the action of G on S has finitely many orbits of two-element subsets of S, and the stabilizer in G of any element of S is finitely generated. Since twisted Brin-Thompson groups are simple, a consequence is that any subgroup of a group admitting an action as above satisfies the Boone-Higman conjecture. In the course of proving this, we also establish a sufficient condition for a group acting cocompactly on a simply connected complex to be finitely presented, even if certain edge stabilizers are not finitely generated, which may be of independent interest.