Twisted Burnside-Frobenius Theorem and $R_{\mathrm{\infty }}$-Property for Lamplighter-Type Groups

We prove that the restricted wreath product ${$\mathbb{Z}$n \mathbin{$\mathrm{w}\mathrm{r}$} $\mathbb{Z}$^k}$hasthe$R\infty$-property, i. e. every its automorphism $\phi$ has infinite Reidemeister number $R(\phi )$, in exactly two cases: (1) for any $k$ and even $n$; (2) for odd $k$ and $n$ divisible by 3. In the remaining cases there are automorphisms with finite Reidemeister number, for which we prove the finite-dimensional twisted Burnside–Frobenius theorem (TBFT): $R(\phi )$ is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action $[\rho ]\mapsto [\rho \circ \phi ]$.