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
Abstract

We construct a family of infinite simple groups that we call twisted Brin–Thompson groups, generalizing Brin’s higher-dimensional Thompson groups $sV$ for $s\in ℕ$. 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 $sV$ 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