Finite presentability of twisted Brin-Thompson groups

TL;DR

This paper characterizes when twisted Brin-Thompson groups are finitely presented, linking group properties and actions, and shows their subgroups satisfy the Boone-Higman conjecture, with broader implications for group theory.

Contribution

It provides a precise criterion for finite presentability of twisted Brin-Thompson groups based on group actions and stabilizers, and introduces a new condition for groups acting on complexes.

Findings

$SV_G$ is finitely presented iff $G$ is finitely presented, has finitely many orbits of pairs, and stabilizers are finitely generated.

Any subgroup of a group with such an action satisfies the Boone-Higman conjecture.

A new sufficient condition for groups acting cocompactly on simply connected complexes to be finitely presented.

Abstract

Given a group $G$ acting faithfully on a set $S$, we characterize precisely when the twisted Brin-Thompson group $SV_G$ is finitely presented. The answer is that $SV_G$ 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.