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

TL;DR

This paper investigates the rationality of the Nielsen zeta function for self-maps on solvmanifolds, proving rationality in low dimensions and under specific algebraic conditions.

Contribution

It extends known results by establishing the rationality of the Nielsen zeta function for self-maps on solvmanifolds of dimension up to 5 and under certain algebraic structures.

Findings

Nielsen zeta function is rational for self-maps of solvmanifolds of dimension ≤ 5.

Rationality holds for self-maps on ${ m NR}$-solvmanifolds.

Rationality also holds for solvmanifolds with fundamental group ${f Z}^n times {f Z}$.

Abstract

In [3,9], 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 $\leq 5$. In any dimension, we show additionally that $N_f(z)$ is rational if $f$ is a self-map of an ${\cal NR}$-solvmanifold or a solvmanifold with fundamental group of the form ${\mathbb Z}^n\rtimes{\mathbb Z}$.