Centralizers in the R. Thompson group $V_n$

Abstract

Let $n\ge 2$ and let $\alpha \in V_n$ be an element in the Higman–Thompson group $V_n$. We study the structure of the centralizer of $\alpha \in V_n$ through a careful analysis of the action of $⟨\alpha ⟩$ on the Cantor set $\mathfrak{C}$. We make use of revealing tree pairs as developed by Brin and Salazar from which we derive discrete train tracks and flow graphs to assist us in our analysis. A consequence of our structure theorem is that element centralizers are finitely generated. Along the way we give a short argument using revealing tree pairs which shows that cyclic groups are undistorted in $V_n$.