Centralizers in the R. Thompson group $V_n$

Let n ≥ 2 and let ∈ Vn be an element in the Higman-Thompson group Vn. We study the structure of the centralizer of ∈ Vn through a careful analysis of the action of hi on the Cantor set 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 Vn.