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.References
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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
- MR Author ID: 54445
- 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
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 2235-2251
- MSC (2000): Primary 22E25, 58F15; Secondary 22D40, 22D45, 05C99
- DOI: https://doi.org/10.1090/S0002-9947-04-03518-4
- MathSciNet review: 2140439