Centralizers in R. Thompson's group V_n

TL;DR

This paper analyzes the structure of centralizers in Higman-Thompson groups V_n, revealing they are finitely generated and providing insights into the group's algebraic properties using tree pair techniques.

Contribution

It introduces a detailed analysis of centralizers in V_n using revealing tree pairs and derives that these centralizers are finitely generated, advancing understanding of the group's structure.

Findings

Centralizers in V_n are finitely generated.

Cyclic groups are undistorted in V_n.

Analysis uses revealing tree pairs and discrete train tracks.

Abstract

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.