Sigma theory and twisted conjugacy, II: Houghton groups and pure symmetric automorphism groups

Daciberg L. Gonçalves and Parameswaran Sankaran

Vol. 280 (2016), No. 2, 349–369

DOI: 10.2140/pjm.2016.280.349

Abstract

Let $\varphi :\Gamma \to \Gamma$ be an automorphism of a group $\Gamma$. We say that $x,y\in \Gamma$ are in the same $\varphi$-twisted conjugacy class and write $x{\sim }_{\varphi }y$ if there exists an element $\gamma \in \Gamma$ such that $y=\gamma x\varphi \left({\gamma }^{-1}\right)$. This is an equivalence relation on $\Gamma$ called the $\varphi$-twisted conjugacy. Let $R\left(\varphi \right)$ denote the number of $\varphi$-twisted conjugacy classes in $\Gamma$. If $R\left(\varphi \right)$ is infinite for all $\varphi \in Aut\left(\Gamma \right)$, we say that $\Gamma$ has the $R_{\infty }$-property.

The purpose of this note is to show that the symmetric group $S_{\infty }$, the Houghton groups and the pure symmetric automorphism groups have the $R_{\infty }$-property. We show, also, that the Richard Thompson group $T$ has the $R_{\infty }$-property. We obtain a general result establishing the $R_{\infty }$-property of the finite direct product of finitely generated groups.

This is a sequel to an earlier work by Gonçalves and Kochloukova, in which it was shown using the sigma theory of Bieri, Neumann and Strebel that, for most of the groups $\Gamma$ considered here, $R\left(\varphi \right)=\infty$ where $\varphi$ varies in a finite index subgroup of the automorphisms of $\Gamma$.

Keywords

twisted conjugacy, Reidemeister number, sigma theory, Houghton groups, infinite symmetric group, pure symmetric automorphism groups

Mathematical Subject Classification 2010

Primary: 20E45

Secondary: 20E36

Milestones

Received: 27 December 2014

Revised: 15 June 2015

Accepted: 6 July 2015

Published: 28 January 2016