A characterization of Anosov rational forms in nilpotent Lie algebras associated to graphs (Q6582381)

It has been conjectured that the only manifolds that can support an Anosov diffeomorphism are those finitely covered by a nilmanifold. A nilmanifold is a manifold that is closely associated with rational nilpotent Lie algebras. In this paper the authors consider existence of Anosov diffeomorphisms for a large class of nilpotent Lie algebras. They look in particular at those that can be realized as a rational form in a Lie algebra associated with a graph.\N\NThe literature in this area has very few examples of real or complex Lie algebras that contain more than one rational form and in which all forms with an Anosov automorphism are classified. In this paper the authors significantly extend the known results by characterizing the existence of Anosov diffeomorphisms on quotients of c-step nilpotent Lie groups associated with graphs.\N\NThe main result provides a class of Lie algebras having distinct rational forms and a general characterization of all Anosov rational forms.