Anosov automorphisms on compact nilmanifolds associated with graphs

Dani, S. G. ; Mainkar, Meera G. (2004) Anosov automorphisms on compact nilmanifolds associated with graphs Transactions of the American Mathematical Society, 357 (6). pp. 2235-2251. ISSN 0002-9947

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Official URL: http://www.ams.org/journals/tran/2005-357-06/S0002...

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.

Item Type:Article
Source:Copyright of this article belongs to American Mathematical Society.
ID Code:74514
Deposited On:16 Dec 2011 09:25
Last Modified:18 May 2016 18:53

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