Reidemeister classes in lamplighter type groups

TL;DR

This paper investigates Reidemeister classes in lamplighter type groups, establishing conditions under which the Reidemeister number is infinite or finite, and confirming the twisted Burnside-Frobenius theorem for certain cases.

Contribution

It characterizes when lamplighter groups have infinite Reidemeister numbers and proves the TBFT for specific classes of these groups.

Findings

Groups $bZ_2 ext{ wr } bZ^k$ and $bZ_3 ext{ wr } bZ^{2d}$ have $R_ty$ property.

Examples of automorphisms with finite Reidemeister numbers in certain groups.

TBFT$_f$ holds for groups with relatively prime conditions on $m$ and 6.

Abstract

We prove that for any automorphism $\phi$ of the restricted wreath product $\mathbb{Z}_2 \mathrm{wr} \mathbb{Z}^k$ and $\mathbb{Z}_3 \mathrm{wr} \mathbb{Z}^{2d}$ the Reidemeister number $R(\phi)$ is infinite, i.e. these groups have the property $R_\infty$. For $\mathbb{Z}_3 \mathrm{wr} \mathbb{Z}^{2d+1}$ and $\mathbb{Z}_p \mathrm{wr} \mathbb{Z}^k$, where $p>3$ is prime, we give examples of automorphisms with finite Reidemeister numbers. So these groups do not have the property $R_\infty$. For these groups and $\mathbb{Z}_m \mathrm{wr} \mathbb{Z}$, where $m$ is relatively prime to $6$, we prove the twisted Burnside-Frobenius theorem (TBFT$_f$): if $R(\phi)<\infty$, then it is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action $[\rho]\mapsto [\rho\circ\phi]$.