Abstract. 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.
Abstract. Let G be a connected Lie group and let F be a lattice in G (not necessarily co-compact). We show that if («,) is a unipotent one-parameter subgroup of G then every ergodic invariant (locally finite) measure of the action of («,) on G/Y is finite. For 'arithmetic lattices' this was proved in [2]. The present generalization is obtained by studying the 'frequency of visiting compact subsets' for unbounded orbits of such flows in the special case where G is a connected semi-simple Lie group of R-rank 1 and Y is any (not necessarily arithmetic) lattice in G.
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