The study of the structure and counting of Reidemeister classes (twisted conjugacy classes) of an automorphism $\phi:G\to G$, i.e. classes $x\sim gx\phi(g^{-1})$, is closely related to the study the twisted inner representation of a discrete group $G$, i.e. a representation on $\ell^2(G)$ corresponding to the action $g\mapsto xg\phi(x^{-1})$ ($x,g\in G$) of $G$ on itself. We study here twisted inner representations from a more general point of view, but the questions under consideration are still close to the important relations to Reidemeister classes. Residually finite groups and amenable groups will be considered in a more detail here.
Joint work with Alexander Fel'shtyn and Nikita Luc