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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2020, Volume 17, Pages 890–898
DOI: https://doi.org/10.33048/semi.2020.17.065
(Mi semr1259)
 

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
References:
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.
Funding agency Grant number
Foundation for the Development of Theoretical Physics and Mathematics BASIS
The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”.
Received May 8, 2020, published July 8, 2020
Bibliographic databases:
Document Type: Article
UDC: 512.547.4, 512.544.43
Language: English
Citation: M. I. Fraiman, “Twisted Burnside–Frobenius theorem and $R_\infty$-property for lamplighter-type groups”, Sib. Èlektron. Mat. Izv., 17 (2020), 890–898
Citation in format AMSBIB
\Bibitem{Fra20}
\by M.~I.~Fraiman
\paper Twisted Burnside--Frobenius theorem and $R_\infty$-property for lamplighter-type groups
\jour Sib. \`Elektron. Mat. Izv.
\yr 2020
\vol 17
\pages 890--898
\mathnet{http://mi.mathnet.ru/semr1259}
\crossref{https://doi.org/10.33048/semi.2020.17.065}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000547538600001}
Linking options:
  • https://www.mathnet.ru/eng/semr1259
  • https://www.mathnet.ru/eng/semr/v17/p890
  • This publication is cited in the following articles:
    1. M. I. Fraiman, V. E. Troitsky, “Reidemeister classes in wreath products of abelian groups”, Sib. elektron. matem. izv., 19:2 (2022), 880–888  mathnet  crossref  mathscinet
    2. E. V. Troitskii, “Reidemeister Classes in Some Lamplighter-Type Groups”, Math. Notes, 113:4 (2023), 605–609  mathnet  crossref  crossref  mathscinet
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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