Geodesic
Reidemeister classes in wreath products of abelian groups
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 2, pp. 880-888.

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Among restricted wreath products $G\wr \mathbb{Z}^k $, where $G$ is a finite abelian group, we find three large classes of groups admitting an automorphism $\varphi$ with finite Reidemeister number $R(\varphi)$ (number of $\varphi$-twisted conjugacy classes). In other words, groups from these classes do not have the $R_\infty$ property. Moreover, we prove that if $\varphi$ is a finite order automorphism of $G\wr \mathbb{Z}^k$ with $R(\varphi)\infty$, then $R(\varphi)$ is equal to the number of fixed points of the map $[\rho]\mapsto [\rho\circ \varphi]$ defined on the set of equivalence classes of finite dimensional irreducible unitary representations of $G\wr \mathbb{Z}^k$.
Keywords: Reidemeister number, twisted conjugacy class, Burnside-Frobenius theorem, unitary dual, finite-dimensional representation. 
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M. I. Fraiman; V. E. Troitsky. Reidemeister classes in wreath products of abelian groups. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 2, pp. 880-888. http://geodesic.mathdoc.fr/item/SEMR_2022_19_2_a11/

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