The Anosov relation for Nielsen numbers of maps of infra-nilmanifolds (Q870656)

The authors prove several theorems on the relation between the Lefschetz number \(L(f)\) and the Nielsen number \(N(f)\) for continuous maps on infra-nilmanifolds. More specifically, let \(M\) be an infra-nilmanifold and \(f:M\to M\) continuous. If the holonomy group of \(M\) is of odd order then \(N(f)=| L(f)| \). Suppose that \(M\) is closed and smooth and \(f\) is a \(C^1\)-map. Then \(f\) is called expanding if for some Riemannian metric there are \(C>0\) and \(\mu>1\) such that \(\| Df^n(v)\| \geq C\mu^n\| v\| \) for all \(v\in TM\). The authors prove that \(N(f)=| L(f)| \) for an expanding map if and only if \(M\) is orientable. An infra-nilmanifold is an orbit space \(M=E\backslash G\) where \(G\) is a connected, simply connected, nilpotent Lie group and \(E\) is a torsion-free almost-crystallographic group. If \(f:M\to M\) is continuous then \(f\) is homotopic to a map which is induced by an affine endomorphism \((\delta,\mathcal{D}):G\to G\). Then \(f\) is called nowhere expanding if all eigenvalues of \(\mathcal{D}_*\) have modulus at most 1. The authors prove that \(N(f)=L(f)\) for nowhere expanding maps \(f:M\to M\).