Which manifolds admit an Anosov diffeomorphism?

Anosov diffeomorphisms are an important class of dynamical systems, which have been intensively studied for more than fifty years now. Next to their dynamical properties, a lot of research deals with the closed manifolds admitting such a diffeomorphism. It is conjectured that up to homeomorphism the only possible manifolds are infra-nilmanifolds, although not every infra-nilmanifold admits an Anosov diffeomorphism. There are two main objectives in our research project to improve the understanding of closed manifolds admitting an Anosov diffeomorphism. The first is to give a new direction towards proving the long-standing conjecture mentioned above. The innovative idea is to translate the notion of an Anosov diffeomorphism to an Anosov automorphism on the fundamental group of the closed manifold. In retrospect, this is exactly how the related result dealing with expanding maps was proved. The second main objective is to give an integrated study of Anosov diffeomorphisms on infra-nilmanifolds. So far, most methods stem from low-dimensional manifolds, making it hard to study Anosov diffeomorphisms in a general way. We hope to give a full characterization of the existence of an Anosov diffeomorphism by the existence of certain gradings which are invariant under the holonomy representation and the action of the Galois group. This method allows us to solve some open problems about the existence of Anosov diffeomorphisms.