Twisted Burnside--Frobenius theorem and $R_\infty$-property for lamplighter-type groups

M. I. Fraiman

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 890-898

Résumé

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 $\varphi$ 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 ${[\rho]\mapsto[\rho\circ\varphi]}$.

Détail

Keywords: Reidemeister number, twisted conjugacy class, Burnside–Frobenius theorem, wreath product.