Anosov Automorphisms on Compact Nilmanifolds Associated with Graphs

We associate with each graph (S, E) a 2-step simply connected nilpotent Lie group N and a lattice Γ in N . We determine the group of Lie automorphisms of N and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold N/Γ 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 there exist a n-dimensional 2-step simply connected nilpotent Lie group N which is indecomposable (not a direct product of lower dimensional nilpotent Lie groups), and a lattice Γ in N such that N/Γ admits an Anosov automorphism; we give also a lower bound on the number of mutually nonisomorphic Lie groups 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.