# Twisted Burnside-Frobenius theory for endomorphisms of polycyclic groups

## TL;DR

This paper extends the Burnside-Frobenius theory to endomorphisms of polycyclic groups, establishing a correspondence between Reidemeister numbers and fixed points of induced maps on the unitary dual, with applications to Gauss congruences.

## Contribution

It proves a twisted Burnside-Frobenius theorem for endomorphisms of polycyclic groups, linking Reidemeister numbers to fixed points on the dual space, and explores finiteness conditions.

## Key findings

- Reidemeister number equals fixed points of induced dual map for certain groups.
- Established Gauss congruences for Reidemeister numbers.
- Identified examples with finite Reidemeister numbers among groups with R_infinity property.

## 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.