Twisted conjugacy classes in unitriangular groups

Let R be an integral domain of characteristic zero. In this note we study the Reidemeister spectrum of the group UTn (R) of unitriangular matrices over R. We prove that if R+ is finitely generated and n > 2 vertical bar R*vertical bar, then UTn (R) possesses the R-infinity-property, i.e. the Reidemeister spectrum of UTn (R) contains only infinity, however, if n <= vertical bar R*vertical bar, then the Reidemeister spectrum of UTn (R) has nonempty intersection with N. If R is a field and n >= 3, then we prove that the Reidemeister spectrum of UTn (R) coincides with (1, infinity}, i.e. in this case UTn (R) does not possess the R-infinity-property.