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

Given an endomorphism φ : G → G on a group G , one can define the Reidemeister number R ( φ ) ∈ N ∪ { ∞ } as the number of twisted conjugacy classes . The corresponding Reidemeister zeta function R φ ( z ) , by using the Reidemeister numbers R ( φ n ) of iterates φ 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 ( φ n ) of all iterates φ 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.