P\'olya-Carlson dichotomy for coincidence Reidemeister zeta functions via profinite completions

TL;DR

This paper studies coincidence Reidemeister zeta functions for nilpotent groups, providing formulas and exploring their analytic properties, including a Pólya-Carlson dichotomy, using profinite completion techniques.

Contribution

It offers a closed formula for coincidence Reidemeister numbers in torsion-free nilpotent groups and investigates the rationality versus natural boundary dichotomy of their zeta functions.

Findings

Derived a closed formula for Reidemeister numbers for certain nilpotent groups.

Established a Pólya-Carlson dichotomy for the zeta functions' analytic behavior.

Applied profinite completion techniques to analyze the zeta functions.

Abstract

We consider coincidence Reidemeister zeta functions for tame endomorphism pairs of nilpotent groups of finite rank, shedding new light on the subject by means of profinite completion techniques. In particular, we provide a closed formula for coincidence Reidemeister numbers for iterations of endomorphism pairs of torsion-free nilpotent groups of finite rank, based on a weak commutativity condition, which derives from simultaneous triangularisability on abelian sections. Furthermore, we present results in support of a P\'olya-Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of the zeta functions in question.