ИСТИНА

The talk is based on the joint works with A. Fel’shtyn, [1–3]. In this talk we will discuss the following statements: 1. the number of twisted conjugacy classes (Reidemeister number) of an automorphism ϕ of a finitely generated residually finite group is equal (if it is finite) to the number of finite dimensional irreducible unitary representations being invariant for the dual of ϕ ; 2. any finitely generated residually finite non-amenable group has the R∞ property (i.e. any automorphism has infinitely many twisted conjugacy classes). Fis gives a lot of new examples and covers many known classes of such groups; Some generalizations and related examples will be discussed, in particular, examples for nonfinitely generated groups. Also we plan to discuss the state of the following our, two year old Conjecture: a finitely generated, residually finite, non-R∞-group is solvable-by-finite.