Anosov automorphisms on compact nilmanifolds associated with graphs

Dani, S.; Mainkar, Meera

doi:10.1090/S0002-9947-04-03518-4

Anosov automorphisms on compact nilmanifolds associated with graphs
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by S. G. Dani and Meera G. Mainkar PDF

Trans. Amer. Math. Soc. 357 (2005), 2235-2251 Request permission

Abstract:

We associate with each graph $(S,E)$ a $2$-step simply connected nilpotent Lie group $N$ and a lattice $\Gamma$ 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 $N/\Gamma$ to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anosov automorphisms, and conclude that for every $n\geq 17$ 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 $\Gamma$ in $N$ such that $N/\Gamma$ 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.

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