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

Abstract

We prove that the restricted wreath product ${\bbz_n \WR \bbz^k}$ has the $R_\infty$-property, i. e. every its automorphism~$\varphi$ has infinite Reidemeister number~$R(\varphi)$, 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 ($\text{TBFT}_f$): $R(\varphi)$ is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action ${[\rho]\mapsto[\rho\circ\varphi]}$.