A note on the $R_\infty$ property for groups $\mathrm{FAlt}(X)\leqslant G\leqslant \mathrm{Sym}(X)$

TL;DR

This paper investigates the $R_ fty$ property in groups related to symmetric groups and Houghton groups, providing conditions under which these groups exhibit this property, especially focusing on infinite, torsion, and residually finite groups.

Contribution

It characterizes the $R_ fty$ property for groups between finitary symmetric groups and full symmetric groups, and extends results to Houghton groups and their commensurable groups.

Findings

Infinite torsion groups between $ ext{FSym}(X)$ and $ ext{Sym}(X)$ have the $R_ fty$ property.

Residually finite groups can act faithfully to produce groups with the $R_ fty$ property.

All groups commensurable to Houghton groups $H_n$ have the $R_ abla$ property.

Abstract

Given a set $X$, the group $\mathrm{Sym}(X)$ consists of all bijections from $X$ to $X$, and $\mathrm{FSym}(X)$ is the subgroup of maps with finite support i.e. those that move only finitely many points in $X$. We describe the automorphism structure of groups $\mathrm{FSym}(X)\le G\le \mathrm{Sym}(X)$ and use this to state some conditions on $G$ for it to have the $R_\infty$ property. Our main results are that if $G$ is infinite, torsion, and $\mathrm{FSym}(X)\le G\le \mathrm{Sym}(X)$, then it has the $R_\infty$ property. Also, if $G$ is infinite and residually finite, then there is a set $X$ such that $G$ acts faithfully on $X$ and, using this action, $\langle G, \mathrm{FSym}(X)\rangle$ has the $R_\infty$ property. Finally we have a result for the Houghton groups, which are a family of groups we denote $H_n$, where $n \in \mathbb{N}$. We show that, given any $n\in \mathbb{N}$, any group commensurable to $H_n$ has the $R_\infty$ property.