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

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 ≤ 5 . In any dimension, we show additionally that N f ( z ) is rational if f is a self-map of an N R -solvmanifold or a solvmanifold with fundamental group of the form Z n ⋊ Z .