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The Nielsen numbers of Anosov diffeomorphisms on flat Riemannian manifolds

Abstract 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 b 1(M) = 0. Since flat manifolds with b 1(M) = 1 or b 1(M) = n − 1 admit no Anosov diffeomorphisms, and the class of flat manifolds with b 1(M) = n consists entirely of tori, a complete picture is obtained.