Generalisations of Thompson's group V arising from purely infinite groupoids

TL;DR

This paper explores generalizations of Thompson's group V derived from purely infinite, minimal groupoids, providing new characterizations and results on their algebraic structure and characters.

Contribution

It introduces a broad class of groups from purely infinite groupoids, characterizes their properties, and describes their derived subgroups and characters.

Findings

Derived subgroup is 2-generated if finitely generated and has no proper characters.

Complete characterization of groups arising from purely infinite, minimal groupoids.

Description of all proper characters of Brin-Higman-Thompson groups.

Abstract

We study a class of generalisations of Thompson's group $V$ arising naturally as topological full groups of purely infinite, minimal groupoids. In the process, we show that the derived subgroup of such a group is 2-generated whenever it is finitely generated and has no proper characters in full generality. We characterise this class of groupoids through a number of group-theoretic conditions on their full groups including vigor, the existence of suitable embeddings of $V$, and compressibility. We moreover give a complete abstract characterisation of those groups that arise as either topological full groups or derived subgroups of purely infinite, minimal groupoids. As an application, we describe all proper characters of the Brin-Higman-Thompson groups.