  For a wide class of groups including polycyclic and finitely generated
polynomial growth groups it is proved that the Reidemeister number of an
automorphism f is equal to the number of finite-dimensional fixed points of the
induced map f^ 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.
