On the rationality of the Nielsen zeta function for maps on solvmanifolds

Abstract

In Dekimpe and Dugardein (J Fixed Point Theory Appl 17:355–370, 2015), Fel’shtyn and Lee (Topol Appl 181:62–103, 2015), the Nielsen zeta function \(N_f(z)\) has been shown to be rational if f is a self-map of an infra-solvmanifold of type (R). It is, however, still unknown whether \(N_f(z)\) is rational for self-maps on solvmanifolds. In this paper, we prove that \(N_f(z)\) is rational if f is a self-map of a (compact) solvmanifold of dimension \(\le 5\). In any dimension, we show additionally that \(N_f(z)\) is rational if f is a self-map of an \(\mathcal{N}\mathcal{R}\)-solvmanifold or a solvmanifold with fundamental group of the form \(\mathbb {Z}^n\rtimes \mathbb {Z}\).