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Centralizers in R.Thompson's group V_n

Bleak, Collin; Bowman, Hannah; Gordon, Alison; Graham, Garrett; Hughes, Jacob; Matucci, Francesco; Sapir, Eugenia

Groups, Geometry and Dynamics, 7(4) (2013), 821–865 (preprint -

Let n be bigger than 1 and let A be an element in the Higman-Thompson group V_n. We study the structure of the centralizer of a in V_n through a careful analysis of the action of the group generated by A 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 to assist us in our analysis. A consequence of our structure theorem is that 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.