Anosov automorphisms on compact nilmanifolds associated with graphs

S. G. Dani, Meera G. Mainkar

Research output: Contribution to journalArticlepeer-review

39 Scopus citations

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.

Original languageEnglish
Pages (from-to)2235-2251
Number of pages17
JournalTransactions of the American Mathematical Society
Volume357
Issue number6
DOIs
StatePublished - Jun 2005

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