Twisted Burnside-Frobenius theory for endomorphisms of polycyclic groups

Abstract

Let $R(\varphi )$ be the number of $\varphi$-conjugacy (or Reidemeister) classes of an endomorphism $\varphi$ of a group $G$. We prove for several classes of groups (including polycyclic) that the number $R(\varphi )$ is equal to the number of fixed points of the induced map of an appropriate subspace of the unitary dual space $\hat{G}$, when $R(\varphi )<\infty$. Applying the result to iterations of $\varphi$ 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.

11 pages, v.2: small corrections, submitted

References in corpus (1)

• Geometry of Reidemeister classes and twisted Burnside theorem

Cited by in corpus (2)

• Dynamical zeta functions of Reidemeister type and representations spaces
• Reidemeister classes in some weakly branch groups