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

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.