## Abstract

For a wide class of groups including polycyclic and finitely generated polynomial growth groups it is proved that the Reidemeister number of an automorphism is equal to the number of finite-dimensional fixed points of the induced map on the unitary dual, if one of these numbers is finite. This theorem is a natural generalization of the classical Burnside-Frobenius theorem to infinite groups. This theorem also has important consequences in topological dynamics and in some sense is a reply to a remark of J.-P. Serre. The main technical results proved in the paper yield a tool for a further progress.

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