Some obstructions on the subgroups of the Brin-Thompson groups and a selection of twisted Brin-Thompson groups

TL;DR

This paper investigates obstructions to subgroups in higher-dimensional and twisted Brin-Thompson groups, revealing similarities to Thompson's group V and excluding certain Baumslag-Solitor groups as subgroups.

Contribution

It extends subgroup obstruction results from Thompson's group V to higher-dimensional and twisted Brin-Thompson groups, identifying key similarities and differences.

Findings

Obstructions on torsion subgroups are similar to those in Thompson's V

Certain Baumslag-Solitor groups are excluded as subgroups

Restrictions on subgroups diminish in the larger class V

Abstract

Motivated by Burillo, Cleary and Roever's summary on obstructions of subgroups of Thompson's group $V,$ we explored the higher dimensional version of the groups, Brin-Thompson groups $nV$ and $SV,$ a class of infinite dimensional Brin-Thompson groups and an easy class of the twisted version of the Brin-Thompson groups $SV_G$ with some certain condition. We found that they have similar obstructions as Thompson's group $V$ on the torsion subgroups and a selection of the interesting Baumslag-Solitor groups are excluded as the subgroups of $SV$ and $SV_G.$ We also discuss a little on the group $\mathscr{S}V,$ an even larger class relaxing some of the "finiteness condition" and observe that some of the restrictions on subgroups disappear.