Twisted conjugacy classes in special and general linear groups

TL;DR

This paper investigates twisted conjugacy classes and the $R_{ty}$-property in classical linear groups, establishing conditions under which these groups exhibit this property.

Contribution

It proves that ${ m GL}_n(K)$ and ${ m SL}_n(K)$ have the $R_{ty}$-property for certain types of integral domains and dimensions.

Findings

${ m GL}_n(K)$ and ${ m SL}_n(K)$ possess $R_{ty}$-property for $n \\geq 3$

The property holds when $K$ has trivial automorphism group

The property also holds when $K$ has zero characteristic and torsion automorphism group

Abstract

In this paper we study twisted conjugacy classes and the $R_{\infty}$-property for classical linear groups. In particular, we prove that the general linear group ${\rm GL}_n(K)$ and the special linear group ${\rm SL}_n(K)$ possess $R_{\infty}$-property, if $n\geq3$ and $K$ is an infinite integral domain with trivial group of automorphisms (Theorem 1), or $K$ is an integral domain, which has a zero characteristic and for which ${\rm Aut}(K)$ is torsion (Theorem 2).