The Anosov theorem for infranilmanifolds with cyclic holonomy group

Abstract

A celebrated theorem of Anosov states that for any continuous self-map f : M → M of a nilmanifold M, the Nielsen number equals the Lefschetz number in absolute value. Anosov also showed that this result does not hold for infranilmanifolds, even in the simplest possible situation of flat manifolds with cyclic holonomy group.Nevertheless, in this paper we extend Anosov's theorem to infranilmanifolds with cyclic holonomy group, provided a certain easily checked condition on the holonomy representation is satisfied.In the case of flat manifolds with cyclic holonomy group this condition is necessary and sufficient. In the general case of all infranilmanifolds with cyclic holonomy group, we provide an example which shows that this condition is no longer necessary.We also prove that for any nonorientable flat manifold Anosov's theorem is not true, but again the same example shows that this is not valid in general for nonorientable infranilmanifolds.

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

“…This is precisely the condition on the holonomy representation of an infra-nilmanifold M with cyclic holonomy group under which the main result of [8] states that the Anosov theorem holds for M . Hence Theorem 3.7 generalises the main result of [8].…”

“…This criterion is a powerful tool in the generalisation of the Anosov theorem to certain classes of maps on infra-nilmanifolds or to certain classes of infra-nilmanifolds. For instance, it lies at the basis of the results in [7] and it is used in [8] to generalise the Anosov theorem to a well described class of infra-nilmanifolds with cyclic holonomy group.…”

Problems on Bieberbach Groups and Flat Manifolds

“…It was recently shown that for large families of flat manifolds (e.g. all flat manifolds with an odd order holonomy group) the Anosov relation N(f ) =| L(f ) | does hold for any self map (see [6] and [7]). It is therefore natural to ask.…”

“…It was first considered by Hantzsche and Wendt. From the other side the fundamental group π 1 (M 3 ) is group F (2,6), where F (r, n) is the group defined by the presentation…”