Reidemeister classes in some wreath products by $\mathbb Z^k$

Abstract

Among restricted wreath products $G\wr \mathbb Z^k $, where $G$ is a finite Abelian group, we find three large classes of groups admitting an automorphism $φ$ with finite Reidemeister number $R(φ)$ (number of $φ$-twisted conjugacy classes). In other words, groups from these classes do not have the $R_\infty$ property. If a general automorphism $φ$ of $G\wr \mathbb Z^k$ has a finite order (this is the case for $φ$ detected in the first part of the paper) and $R(φ)<\infty$, we prove that $R(φ)$ coincides with the number of equivalence classes of finite-dimensional irreducible unitary representations of $G\wr \mathbb Z^k$, which are fixed by the dual map $[ρ]\mapsto [ρ\circ φ]$ (i.e. we prove the conjecture about finite twisted Burnside-Frobenius theorem, TBFT$_f$, for these $φ$).