This leads of course to the question how the structure of the universal covering group influences the validity of the Anosov theorem?In Part 3 we present for each (almost-)Bieberbach group a proof of, or a counter example to, the Anosov relation for this specific (almost-) Bieberbach group (or infra-nilmanifold). As we already argued above, an analysis of the obtained results can hopefully form the basis for future research.

An a½ne di¨eomorphism of an infra-nilmanifold M (with universal cover L) is virtually unipotent if and only if it lifts to an a½ne automorphism of L whose linear part has only eigenvalues of absolute value one. A continuous self-map of M which is homotopic to a virtually unipotent a½ne di¨eomorphism of M is referred to as a virtually unipotent map of M. In particular, we note that homotopically periodic self-maps of M are virtually unipotent.The main result of this paper is that, for each virtually unipotent map f of an infranilmanifold M, the Lefschetz number L f and the Nielsen number N f coincide. This nicely generalizes the analogous result of S. Kwasik and K. B. Lee, as presented in [11], for homotopically periodic maps of infra-nilmanifolds.

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