Twisted conjugacy classes in unitriangular groups

Abstract

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

arXiv commentSome mistake corrected in the second (present) version