Geodesic
Twisted Burnside--Frobenius theorem and $R_\infty$-property for lamplighter-type groups
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 890-898.

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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. 
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     author = {M. I. Fraiman},
     title = {Twisted {Burnside--Frobenius} theorem and $R_\infty$-property for lamplighter-type groups},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
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     volume = {17},
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     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a16/}
}
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M. I. Fraiman. Twisted Burnside--Frobenius theorem and $R_\infty$-property for lamplighter-type groups. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 890-898. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a16/

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