Volume 26, issue 3 (2022)

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Twisted Brin–Thompson groups

James Belk and Matthew C B Zaremsky

Geometry & Topology 26 (2022) 1189–1223

We construct a family of infinite simple groups that we call twisted Brin–Thompson groups, generalizing Brin’s higher-dimensional Thompson groups sV for s . 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 sV and hence every right-angled Artin group, including examples of type  F and a family of examples of type F n1 but not of type  F n for arbitrary n . This provides the second known infinite family of simple groups distinguished by their finiteness properties.

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Thompson group, finiteness properties, simple group, right-angled Artin group, quasi-isometry, oligomorphic, Cantor space
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20E32, 57M07
Received: 13 February 2020
Accepted: 1 May 2021
Published: 3 August 2022
Proposed: Martin R Bridson
Seconded: Mladen Bestvina, David M Fisher
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