Presentations of generalisations of Thompson's group $V$

TL;DR

This paper studies generalizations of Thompson's group V, showing they are finitely generated and finitely presented under certain conditions, and provides methods for explicit presentations and their simplification.

Contribution

The authors extend previous results to show that groups V_r(Σ) are finitely generated and presented, and develop explicit presentation and simplification techniques.

Findings

V_r(Σ) groups are finitely generated under mild conditions.

Explicit presentations for these groups are constructed.

A general procedure for shortening presentations is provided.

Abstract

We consider generalisations of Thompson's group $V$, denoted by $V_r(\Sigma)$, which also include the groups of Higman, Stein and Brin. It was shown by the authors in [20] that under some mild conditions these groups and centralisers of their finite subgroups are of type $\mathrm{F}_\infty$. Under more general conditions we show that the groups $V_r(\Sigma)$ are finitely generated and, under the mild conditions mentioned above, we see that they are finitely presented and give a recipe to find explicit presentations. For the centralisers of finite subgroups we find a suitable infinite presentation and then apply a general procedure to shorten this presentation. In the appendix, we give a proof of this general shortening procedure.