Twisted Brin–Thompson groups

We construct a family of infinite simple groups that we call twisted Brin–Thompson groups, generalizing Brin’s higher-dimensional Thompson groups $s V$ for $s \in ℕ$ . We use twisted Brin–Thompson groups to prove a variety of results regarding simple groups. For example, we prove that every finitely generated group embeds quasi-isometrically as a subgroup of a two-generated simple group, strengthening a result of Bridson. We also produce examples of simple groups that contain every $s V$ and hence every right-angled Artin group, including examples of type $F_{\infty}$ and a family of examples of type $F_{n - 1}$ but not of type $F_{n}$ for arbitrary $n \in ℕ$ . This provides the second known infinite family of simple groups distinguished by their finiteness properties.

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Keywords

Thompson group, finiteness properties, simple group, right-angled Artin group, quasi-isometry, oligomorphic, Cantor space

Mathematical Subject Classification 2010

Primary: 20F65

Secondary: 20E32, 57M07

References

Publication

Received: 13 February 2020

Accepted: 1 May 2021

Published: 3 August 2022

Proposed: Martin R Bridson

Seconded: Mladen Bestvina, David M Fisher

Authors

James Belk
School of Mathematics and Statistics University of St Andrews St Andrews United Kingdom

Matthew C B Zaremsky
Department of Mathematics and Statistics University at Albany (SUNY) Albany, NY United States

Volume 26, issue 3 (2022)

James Belk and Matthew C B Zaremsky

Abstract