Twisted Brin-Thompson groups

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 ${\text{F}}_{\infty }$ and a family of examples of type ${\text{F}}_{n-1}$ but not of type ${\text{F}}_n$, for arbitrary $n\in \mathbb{N}$. This provides the second known infinite family of simple groups distinguished by their finiteness properties.

26 pages, 3 figures. v2: final version, to appear in Geometry & Topology

References in corpus (2)

• Classifying spaces from Ore categories with Garside families
• Sufficient conditions for a group of homeomorphisms of the Cantor set to be two-generated

Cited by in corpus (5)

• Asymptotically rigid mapping class groups I: Finiteness properties of braided Thompson's and Houghton's groups
• Embeddings into left-orderable simple groups
• Sufficient conditions for a group of homeomorphisms of the Cantor set to be two-generated
• Finiteness properties for relatives of braided Higman--Thompson groups
• Asymptotically rigid mapping class groups II: strand diagrams and nonpositive curvature