Twisted conjugacy classes in unitriangular groups

Abstract

Abstract Let R be an integral domain of characteristic zero. In this note we study the Reidemeister spectrum of the group UT n ⁢ ( R ) {{\rm UT}_{n}(R)} of unitriangular matrices over R. We prove that if R + {R^{+}} is finitely generated and n > 2 ⁢ | R * | {n>2|R^{*}|} , then UT n ⁢ ( R ) {{\rm UT}_{n}(R)} possesses the R ∞ {R_{\infty}} -property, i.e. the Reidemeister spectrum of UT n ⁢ ( R ) {{\rm UT}_{n}(R)} contains only ∞ {\infty} , however, if n ≤ | R * | {n\leq|R^{*}|} , then the Reidemeister spectrum of UT n ⁢ ( R ) {{\rm UT}_{n}(R)} has nonempty intersection with ℕ {\mathbb{N}} . If R is a field and n ≥ 3 {n\geq 3} , then we prove that the Reidemeister spectrum of UT n ⁢ ( R ) {{\rm UT}_{n}(R)} coincides with { 1 , ∞ } {\{1,\infty\}} , i.e. in this case UT n ⁢ ( R ) {{\rm UT}_{n}(R)} does not possess the R ∞ {R_{\infty}} -property.