The Nielsen Numbers of Homotopically Periodic Maps of Infranilmanifolds

Let f : M → M be a homotopically periodic (that is, f k ≃ id, for some k ⩾ 1) self‐map of an infranilmanifold M . Then the Lefschetz number L ( f ) and the Nielsen number N ( f ) of the map f are the same. In particular, L ( f ) = N ( f ) for f an isometry. On the other hand, contrary to the Anosov result for nilmanifolds, | L ( f )| ≃ N ( f ) even for f an affine Anosov diffeomorphism. However, the deviation from the equality | L ( f )| = N ( f ) can be measured by the holonomy group of M .