Twisted conjugacy classes in unitriangular groups
Timur Nasybullov

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 be an integral domain of zero characteristic. In this note we study the Reidemeister spectrum of the group of unitriangular matrices over . We prove that if is finitely generated and , then possesses the -property, i. e. the Reidemeister spectrum of contains only , however, if , then the Reidemeister spectrum of has nonempty intersection with . If is a field, then we prove that the Reidemeister spectrum of coincides with , i. e. in this case does not possess the -property.
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