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 Fundam. Prikl. Mat., 2016, Volume 21, Issue 5, Pages 219–227 (Mi fpm1765)

Two examples related to the twisted Burnside–Frobenius theory for infinitely generated groups

E. V. Troitskii

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The TBFT$_f$ conjecture, which is a modification of a conjecture by Fel'shtyn and Hill, says that if the Reidemeister number $R(\phi)$ of an automorphism $\phi$ of a (countable discrete) group $G$ is finite, then it coincides with the number of fixed points of the corresponding homeomorphism $\hat{\phi}$ of $\hat{G}_f$ (the part of the unitary dual formed by finite-dimensional representations). The study of this problem for residually finite groups has been the subject of some recent activity. We prove here that for infinitely generated residually finite groups there are positive and negative examples for this conjecture. It is detected that the finiteness properties of the number of fixed points of $\phi$ itself also differ from the finitely generated case.

 Funding Agency Grant Number Russian Science Foundation 16-11-10018

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Citation: E. V. Troitskii, “Two examples related to the twisted Burnside–Frobenius theory for infinitely generated groups”, Fundam. Prikl. Mat., 21:5 (2016), 219–227

Citation in format AMSBIB
\Bibitem{Tro16} \by E.~V.~Troitskii \paper Two examples related to the twisted Burnside--Frobenius theory for infinitely generated groups \jour Fundam. Prikl. Mat. \yr 2016 \vol 21 \issue 5 \pages 219--227 \mathnet{http://mi.mathnet.ru/fpm1765}