Centralizers in the R. Thompson group $V_n$

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

doi:10.4171/ggd/207

JournalsggdVol. 7, No. 4pp. 821–865

Centralizers in the R. Thompson group $V_{n}$

Collin Bleak
University of St Andrews, United Kingdom
Hannah Bowman
New York, USA
Alison Gordon Lynch
University of Wisconsin, Madison, USA
Garrett Graham
University of California, San Diego, La Jolla, USA
Jacob Hughes
University of California, San Diego, La Jolla, USA
Francesco Matucci
Université Paris-Sud, Orsay, France
Eugenia Sapir
Stanford University, USA

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Abstract

Let $n \geq 2$ and let $α \in V_{n}$ be an element in the Higman–Thompson group $V_{n}$ . We study the structure of the centralizer of $α \in V_{n}$ through a careful analysis of the action of $⟨ α ⟩$ 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 $V_{n}$ .

Cite this article

Collin Bleak, Hannah Bowman, Alison Gordon Lynch, Garrett Graham, Jacob Hughes, Francesco Matucci, Eugenia Sapir, Centralizers in the R. Thompson group $V_{n}$ . Groups Geom. Dyn. 7 (2013), no. 4, pp. 821–865

DOI 10.4171/GGD/207