The Higman--Thompson groups $V_n$ are $(2,2,2)$-generated

TL;DR

This paper demonstrates that Higman--Thompson groups $V_n$ can be generated by three involutions, providing a new family of generating sets parametrized by sequences in $V_n$, with implications for understanding their algebraic structure.

Contribution

Introduces a family of generating sets for $V_n$ consisting of three involutions, inspired by spinal elements in branch group theory, showing $V_n$ is $(2,2,2)$-generated.

Findings

Existence of generating sets of $V_n$ with three involutions

Parametrization of generating sets by sequences in $V_n$

Connection to spinal elements in branch groups

Abstract

We provide a family of generating sets $S_{\alpha}$ of the Higman--Thompson groups $V_n$ that are parametrized by certain sequences $\alpha$ of elements in $V_n$. These generating sets consist of $3$ involutions $\sigma$, $\tau$, and $s_{\alpha}$, where the latter involution is inspired by the class of spinal elements in the theory of branch groups. In particular this shows the existence of generating sets of $V_n$ that consist of $3$ involutions.