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This article is cited in 2 scientific papers (total in 2 papers)
Mathematical logic, algebra and number theory
Twisted Burnside–Frobenius theorem and $R_\infty$-property for lamplighter-type groups
M. I. Fraimanab a 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.
Received May 8, 2020, published July 8, 2020
Citation:
M. I. Fraiman, “Twisted Burnside–Frobenius theorem and $R_\infty$-property for lamplighter-type groups”, Sib. Èlektron. Mat. Izv., 17 (2020), 890–898
Linking options:
https://www.mathnet.ru/eng/semr1259 https://www.mathnet.ru/eng/semr/v17/p890
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