Anosov automorphisms on compact nilmanifolds associated with graphs
Authors:
S. G. Dani and Meera G. Mainkar
Journal:
Trans. Amer. Math. Soc. 357 (2005), 22352251
MSC (2000):
Primary 22E25, 58F15; Secondary 22D40, 22D45, 05C99
Published electronically:
April 27, 2004
MathSciNet review:
2140439
Fulltext PDF Free Access
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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:
http://dx.doi.org/10.1090/S0002994704035184
PII:
S 00029947(04)035184
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 secondnamed author gratefully acknowledges partial support from the TIFR Alumni Association Scholarship of the TIFR Endowment Fund
Article copyright:
© Copyright 2004
American Mathematical Society
