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

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.