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

DOI:
https://doi.org/10.1090/S0002-9947-04-03518-4

Published electronically:
April 27, 2004

MathSciNet review:
2140439

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Abstract: We associate with each graph a -step simply connected nilpotent Lie group and a lattice in . We determine the group of Lie automorphisms of and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anosov automorphisms, and conclude that for every there exist a -dimensional -step simply connected nilpotent Lie group which is indecomposable (not a direct product of lower dimensional nilpotent Lie groups), and a lattice in such that admits an Anosov automorphism; we give also a lower bound on the number of mutually nonisomorphic Lie groups 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:
https://doi.org/10.1090/S0002-9947-04-03518-4

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