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 \langle \alpha \rangle 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 .