Reidemeister classes in wreath products of abelian groups (Q6642690)

Let \(G\) be a group and \(\phi \in \mathrm{Aut}(G)\). Two elements \(x,y \in G\) are called \(\phi\)-conjugate (or twisted conjugate) in \(G\) if there exists \(g\in G\) such that \(y=g^{-1}xg^{\phi}\). The corresponding classes are called Reidemeister or twisted conjugacy classes and the number \(R(\phi)\) of them is called the Reidemeister number of \(\phi\). A group \(G\) has \(R_{\infty}\) property if \(R(\phi)=\infty\) for every \(\phi \in \mathrm{Aut}(G)\).\N\NThe authors, among restricted wreath products \(G\wr \mathbb{Z}^{k}\), where \(G\) is a finite abelian group, find three large classes of groups admitting an automorphism \(\phi\) with finite Reidemeister number \(R(\phi)\), so groups from these classes do not have the \(R_{\infty}\) property. Moreover, they prove that if \(\phi\) is an automorphism of finite order of \(G \wr \mathbb{Z}^{k}\) with \(R(\phi) \leq \infty\), then \(R(\phi)\) is equal to the number of fixed points of the map \([\rho] \mapsto [\rho \circ \phi]\) defined on the set of equivalence classes of finite dimensional irreducible unitary representations of \(G \wr \mathbb{Z}^{k}\).