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

Abstract

We prove that the restricted wreath product ${\mathbb{Z}}_n\mathrm{wr}{\mathbb{Z}}^k$ has the $R_{\infty }$-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 $[\rho ]\mapsto [\rho \circ \phi ]$.

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• Twisted conjugacy in some lamplighter-type groups