A note on the existence of the Reidemeister zeta function on groups

Abstract

Given an endomorphism $\varphi: G \to G$ on a group $G$, one can define the Reidemeister number $R(\varphi) \in \mathbb{N} \cup \{\infty\}$ as the number of twisted conjugacy classes. The corresponding Reidemeister zeta function $R_\varphi(z)$, by using the Reidemeister numbers $R(\varphi^n)$ of iterates $\varphi^n$ in order to define a power series, has been studied a lot in the literature, especially the question whether it is a rational function or not. For example, it has been shown that the answer is positive for finitely generated torsion-free virtually nilpotent groups, but negative in general for abelian groups that are not finitely generated. However, in order to define the Reidemeister zeta function of an endomorphism $\varphi$, it is necessary that the Reidemeister numbers $R(\varphi^n)$ of all iterates $\varphi^n$ are finite. This puts restrictions, not only on the endomorphism $\varphi$, but also on the possible groups $G$ if $\varphi$ is assumed to be injective. In this note, we want to initiate the study of groups having a well-defined Reidemeister zeta function for a monomorphism $\varphi$, because of its importance for describing the behavior of Reidemeister zeta functions. As a motivational example, we show that the Reidemeister zeta function is indeed rational on torsion-free virtually polycyclic groups. Finally, we give some partial results about the existence in the special case of automorphisms on finitely generated torsion-free nilpotent groups, showing that it is a restrictive condition.