Work

Twisted Brin–Thompson groups

Year: 2022

Type: article

Abstract: We construct a family of infinite simple groups that we call \emph{twisted Brin-Thompson groups}, generalizing Brin's higher-dimensional Thompson groups $sV$ ($s\in\mathbb{N}$). 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 $\textrm{F}_\infty$ and a family of examples of type $\textrm{F}_{n-1}$ but not of type $\textrm{F}_n$, for arbitrary $n\in\mathbb{N}$. This provides the second known infinite family of simple groups distinguished by their finiteness properties.

Source: Geometry & Topology

DOI: 10.2140/gt.2022.26.1189

Authors: James Belk, Matthew C B Zaremsky

Institutions: University of St Andrews, University at Albany, State University of New York

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Engineering and Physical Sciences Research Council

Awards: G8631404110, Bi-synchronizing automata, outer automorphism groups of Higman-Thompson groups, and automorphisms of the shift.