This article is cited in 2 papers

Mathematical logic, algebra and number theory

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

M. I. Fraiman<sup>ab

a</sup> Faculty of Mechanics and Mathematics, Lomonosov Moscow State University
^b Moscow Center for Fundamental and Applied Mathematics, MSU Department, 1, Leninskiye Gory st., Moscow, 119991, Russia

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

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

UDC: 512.547.4, 512.544.43

MSC: 20E22, 20E36, 20E45, 22D10

Received May 8, 2020, published July 8, 2020

Language: English

DOI: 10.33048/semi.2020.17.065

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