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

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.