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Karel Dekimpe, Bram De Rock, Wim Malfait

The Nielsen numbers of Anosov diffeomorphisms on flat Riemannian manifolds

In this paper we study the relation between the Lefschetz number and the Nielsen number of an Anosov diffeomorphism on a flat manifold. As a first result we obtain that for each n ? 4 and each k satisfying 2 ? k ? n ? 2, there exists a flat n-dimensional manifold M having first Betti number b₁(M) = k and admitting an Anosov diffeomorphism f on M with N(f) ? |L(f )|. On the other hand, in almost all cases one can also construct on the same manifold M an Anosov diffeomorphism g with N(g) = |L(g)|. Analogous results are obtained in the case of primitive flat manifolds M, i.e. with b₁(M) = 0. Since flat manifolds with b₁(M) = 1 or b₁(M) = n ? 1 admit no Anosov diffeomorphisms, and the class of flat manifolds with b₁(M) = n consists entirely of tori, a complete picture is obtained.

Forum Mathematicum, Walter de Gruyter

Print ISSN: 0933-7741
Volume: 17, 03/2005
Pages: 325 - 341

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