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

Abstract

We provide a family of generating sets $S_{\hat{I}}\pm$ of the Higman--Thompson groups $V_n$ that are parametrized by certain sequences $\hat{I}\pm$ of elements in $V_n$. These generating sets consist of $3$ involutions $\ddot{I}ƒ$, $\ddot{I}„$, and $s_{\hat{I}}\pm$, 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.

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