Twisted conjugacy classes in unitriangular groups

TL;DR

This paper investigates the Reidemeister spectrum of unitriangular groups over integral domains, establishing conditions under which these groups have the $R_{inite}$-property or spectrum containing finite numbers.

Contribution

It provides new results on the Reidemeister spectrum of ${ m UT}_n(R)$, including criteria for the $R_{inite}$-property based on the size of $n$ relative to the units in $R$.

Findings

If $R^+$ is finitely generated and $n>2|R^*|$, then ${ m UT}_n(R)$ has the $R_{inite}$-property.

If $n leq |R^*|$, the spectrum contains only $ ext{infinity}$.

For fields $R$, the spectrum is exactly $oxed{ ext{1, infinity}}$.

Abstract

Let $R$ be an integral domain of zero characteristic. In this note we study the Reidemeister spectrum of the group ${\rm UT}_n(R)$ of unitriangular matrices over $R$. We prove that if $R^+$ is finitely generated and $n>2|R^*|$, then ${\rm UT}_n(R)$ possesses the $R_{\infty}$-property, i. e. the Reidemeister spectrum of ${\rm UT}_n(R)$ contains only $\infty$, however, if $n\leq|R^*|$, then the Reidemeister spectrum of ${\rm UT}_n(R)$ has nonempty intersection with $\mathbb{N}$. If $R$ is a field, then we prove that the Reidemeister spectrum of ${\rm UT}_n(R)$ coincides with $\{1,\infty\}$, i. e. in this case ${\rm UT}_n(R)$ does not possess the $R_{\infty}$-property.