Twisted Burnside-Frobenius theory for endomorphisms of polycyclic groups

Abstract

Let $R(\phi)$ be the number of $\phi$-conjugacy (or Reidemeister) classes of an endomorphism $\phi$ of a group $G$. We prove for several classes of groups (including polycyclic) that the number $R(\phi)$ is equal to the number of fixed points of the induced map of an appropriate subspace of the unitary dual space $\widehat G$, when $R(\phi)<\infty$. Applying the result to iterations of $\phi$ we obtain Gauss congruences for Reidemeister numbers. In contrast with the case of automorphisms, studied previously, we have a plenty of examples having the above finiteness condition, even among groups with $R_\infty$ property.Comment: 11 pages, v.2: small corrections, submitte