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

TL;DR

This paper proves that certain infinite groups, including Houghton groups, pure symmetric automorphism groups, and the Richard Thompson group, possess the $R_$-property, meaning all their automorphisms have infinitely many twisted conjugacy classes.

Contribution

It establishes the $R_$-property for several important classes of groups, including Houghton groups, pure symmetric automorphism groups, and the Thompson group T, and provides a general result for finite direct products.

Findings

Houghton groups have the $R_$-property.

Pure symmetric automorphism groups have the $R_$-property.

The Thompson group T has the $R_$-property.

Abstract

We say that $x,y\in \Gamma$ are in the same $\phi$-twisted conjugacy class and write $x\sim_\phi y$ if there exists an element $\gamma\in \Gamma$ such that $y=\gamma x\phi(\gamma^{-1})$. This is an equivalence relation on $\Gamma$ called the $\phi$-twisted conjugacy. Let $R(\phi)$ denote the number of $\phi$-twisted conjugacy classes in $\Gamma$. If $R(\phi)$ is infinite for all $\phi\in Aut(\Gamma)$, 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 finite direct product of finitely generated groups.