Groups, Geometry, and Dynamics

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Volume 7, Issue 4, 2013, pp. 821–865
DOI: 10.4171/GGD/207

Published online: 2013-11-20

Centralizers in the R. Thompson group $V_n$

Collin Bleak[1], Hannah Bowman[2], Alison Gordon Lynch[3], Garrett Graham[4], Jacob Hughes[5], Francesco Matucci[6] and Eugenia Sapir[7]

(1) University of St Andrews, United Kingdom
(2) New York, USA
(3) University of Wisconsin, Madison, USA
(4) University of California, San Diego, La Jolla, USA
(5) University of California, San Diego, La Jolla, USA
(6) Université Paris-Sud, Orsay, France
(7) Stanford University, USA

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 $V_n$.

Keywords: Conjugacy, centralizer, Thompson's group $V$, train track, flow graph

Bleak Collin, Bowman Hannah, Gordon Lynch Alison, Graham Garrett, Hughes Jacob, Matucci Francesco, Sapir Eugenia: Centralizers in the R. Thompson group $V_n$. Groups Geom. Dyn. 7 (2013), 821-865. doi: 10.4171/GGD/207