The Anosov Theorem for Infra-Nilmanifolds with a 2-Perfect Holonomy Group

In this paper, we show that $N(f) = |L(f)|$ for any continuous selfmap $f : M → M$ on an infra-nilmanifold $M$ of which the holonomy group is 2-perfect (i.e. having no index two subgroup). Conversely, for any finite group $F$ that is not 2-perfect, we show there exists at least one infra-nilmanifold $M$ with holonomy group $F$ and a continuous selfmap $f : M → M$ such that $N(f) \neq |L(f)|$.

Keywords

Nielsen number, Lefschetz number, infra-nilmanifold, holonomy group

2010 Mathematics Subject Classification

37C25, 54H25, 55M20

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Published 17 April 2012