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

Abstract

Given an endomorphism $\phi :G\to G$ on a group G, one can define the Reidemeister number $R\left(\phi \right)\in N\cup \{\infty \}$ as the number of twisted conjugacy classes. The corresponding Reidemeister zeta function $R_{\phi }\left(z\right)$, by using the Reidemeister numbers $R\left({\phi }^n\right)$ of iterates ${\phi }^n$ 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 φ, it is necessary that the Reidemeister numbers $R\left({\phi }^n\right)$ of all iterates ${\phi }^n$ are finite. This puts restrictions, not only on the endomorphism φ, but also on the possible groups G if φ 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 φ, 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.