Towards a dichotomy for the Reidemeister zeta function

TL;DR

This paper establishes a clear dichotomy in the analytic behavior of the Reidemeister zeta function, showing it is either rational or has a natural boundary, for various classes of groups and maps.

Contribution

It introduces a novel dichotomy for the Reidemeister zeta function's behavior across different group types and maps, expanding understanding of its analytic properties.

Findings

Dichotomy between rationality and natural boundary for the Reidemeister zeta function.

Rationality of the coincidence Reidemeister zeta function for certain nilpotent groups.

Application of the dichotomy to topological spaces with non-finitely generated torsion abelian fundamental groups.

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.