Towards a dichotomy for the Reidemeister zeta function

Abstract

We prove a dichotomy between rationality and a natural boundary for the analytic behavior of the Reidemeister zeta function for automorphisms of non-finitely generated torsion abelian groups and for endomorphisms of groups ${\mathbb{Z}}_p^d,$ where ${\mathbb{Z}}_p$ the group of p-adic integers. As a consequence, we obtain a dichotomy for the Reidemeister zeta function of a continuous map of a topological space with fundamental group that is non-finitely generated torsion abelian group. We also prove the rationality of the coincidence Reidemeister zeta function for tame endomorphisms pairs of finitely generated torsion-free nilpotent groups, based on a weak commutativity condition.

25 pages