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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Anosov automorphisms on compact nilmanifolds associated with graphs


Authors: S. G. Dani and Meera G. Mainkar
Journal: Trans. Amer. Math. Soc. 357 (2005), 2235-2251
MSC (2000): Primary 22E25, 58F15; Secondary 22D40, 22D45, 05C99
Published electronically: April 27, 2004
MathSciNet review: 2140439
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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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Additional Information

S. G. Dani
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Bombay 400 005, India
Email: dani@math.tifr.res.in

Meera G. Mainkar
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Bombay 400 005, India
Email: meera@math.tifr.res.in

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03518-4
PII: S 0002-9947(04)03518-4
Received by editor(s): February 28, 2003
Received by editor(s) in revised form: July 16, 2003
Published electronically: April 27, 2004
Additional Notes: The second-named author gratefully acknowledges partial support from the TIFR Alumni Association Scholarship of the TIFR Endowment Fund
Article copyright: © Copyright 2004 American Mathematical Society