A taste of twisted Brin-Thompson groups

TL;DR

This paper introduces twisted Brin-Thompson groups, highlighting their ability to embed various groups into simple groups and constructing simple groups with arbitrary finiteness length, with implications for the Boone-Higman Conjecture.

Contribution

It provides a concise introduction to twisted Brin-Thompson groups and proves a new embedding result for finitely presented groups with specific actions.

Findings

Finitely generated groups quasi-isometrically embed into finitely generated simple groups.

Constructs simple groups with any given finiteness length.

Embeds certain finitely presented groups into finitely presented simple groups.

Abstract

This note serves as a short and reader-friendly introduction to twisted Brin-Thompson groups, which were recently constructed by Belk and the author to provide a family of simple groups with a variety of interesting properties. Most notably, twisted Brin-Thompson groups can be used to show that every finitely generated group quasi-isometrically embeds as a subgroup of a finitely generated simple group. Another important application is a concrete construction of a family of simple groups with arbitrary finiteness length. In addition to giving a concise introduction to the groups and these applications, we also prove here a strengthening of one of the results from the original paper. Namely, we prove that any finitely presented group acting faithfully and oligomorphically on a set, with finitely generated stabilizers of finite subsets, embeds in a finitely presented simple group. We believe this could potentially lead to future progress on resolving the Boone-Higman Conjecture for certain groups.