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 $\phi$ with finite Reidemeister number $R(\phi )$ (number of $\phi$-twisted conjugacy classes). In other words, groups from these classes do not have the $R_{\infty }$ property. If a general automorphism $\phi$ of $G\wr {\mathbb{Z}}^k$ has a finite order (this is the case for $\phi$ detected in the first part of the paper) and $R(\phi )<\infty$, we prove that $R(\phi )$ 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 $[\rho ]\mapsto [\rho \circ \phi ]$ (i.e. we prove the conjecture about finite twisted Burnside-Frobenius theorem, TBFT${}_f$, for these $\phi$).