Reidemeister classes in lamplighter-type groups

Abstract

We prove that for any automorphism $\varphi$ of the restricted wreath product ${\mathbb{Z}}_2\text{wr}{\mathbb{Z}}^k$ and ${\mathbb{Z}}_3\text{wr}{\mathbb{Z}}^{2d}$ the Reidemeister number $R(\varphi )$ is infinite (the property $R_{\infty }$). For ${\mathbb{Z}}_3\text{wr}{\mathbb{Z}}^{2d+1}$ and ${\mathbb{Z}}_p\text{wr}{\mathbb{Z}}^k$, 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 ${\mathbb{Z}}_m\text{wr}\mathbb{Z}$, where m is relatively prime to 6, we prove the twisted Burnside-Frobenius theorem (TBFT_f): if $R(\varphi )<\infty$, then it is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action $[\rho ]\mapsto [\rho °\varphi ]$.

Keywords:

• Reidemeister number
• $R_{\infty }$-group
• twisted conjugacy class
• residually finite group
• restricted wreath product
• lamplighter group

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 20C
• 20E45
• 20E22
• 20E26
• 20F16
• 22D10

Acknowledgements

The author is indebted to A. Fel’shtyn for helpful discussions in the Max-Planck Institute for Mathematics (Bonn) in February, 2017 and the MPIM for supporting this visit. The author is grateful to L. Alania, R. Jimenez Benitez, and V. Manuilov for valuable advises and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work is supported by the Russian Science Foundation under grant 16-11-10018.