Centralizers in the R. Thompson group [V.sub.n]

Let n [greater than or equal to] 2 and let [alpha] [member of] [V.sub.n] be an element in the Higman-Thompson group [V.sub.n]. We study the structure of the centralizer of a 2 Vn through a careful analysis of the action of 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 [V.sub.n]. Mathematics Subject Classification (2010). 20F65, 20E07, 37C85. Keywords. Conjugacy, centralizer, Thompson's group V, train track, flow graph.