Anosov automorphisms on compact nilmanifolds associated with graphs

Abstract

We associate with each graph ( S , E ) (S,E) a 2 2 -step simply connected nilpotent Lie group N N and a lattice Γ \Gamma in N N . We determine the group of Lie automorphisms of N N and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold N / Γ N/\Gamma to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anosov automorphisms, and conclude that for every n ≥ 17 n\geq 17 there exist a n n -dimensional 2 2 -step simply connected nilpotent Lie group N N which is indecomposable (not a direct product of lower dimensional nilpotent Lie groups), and a lattice Γ \Gamma in N N such that N / Γ N/\Gamma admits an Anosov automorphism; we give also a lower bound on the number of mutually nonisomorphic Lie groups N N of a given dimension, satisfying the condition. Necessary and sufficient conditions are also described for a compact nilmanifold as above to admit ergodic automorphisms.