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

 

Nilmanifolds of dimension $ \leq 8$ admitting Anosov diffeomorphisms


Authors: Jorge Lauret and Cynthia E. Will
Journal: Trans. Amer. Math. Soc. 361 (2009), 2377-2395
MSC (2000): Primary 37D20; Secondary 22E25, 20F34
Published electronically: November 25, 2008
MathSciNet review: 2471923
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Abstract: After more than thirty years, the only known examples of Anosov diffeomorphisms are topologically conjugated to hyperbolic automorphisms of infranilmanifolds, and even the existence of an Anosov automorphism is a really strong condition on an infranilmanifold. Any Anosov automorphism determines an automorphism of the rational Lie algebra determined by the lattice, which is hyperbolic and unimodular (and conversely ...). These two conditions together are strong enough to make of such rational nilpotent Lie algebras (called Anosov Lie algebras) very distinguished objects. In this paper, we classify Anosov Lie algebras of dimension less than or equal to 8.

As a corollary, we obtain that if an infranilmanifold of dimension $ n\leq 8$ admits an Anosov diffeomorphism $ f$ and it is not a torus or a compact flat manifold (i.e. covered by a torus), then $ n=6$ or 8 and the signature of $ f$ necessarily equals $ \{ 3,3\}$ or $ \{ 4,4\}$, respectively.


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Additional Information

Jorge Lauret
Affiliation: FaMAF and CIEM, Universidad Nacional de Córdoba, Córdoba, Argentina
Email: lauret@mate.uncor.edu

Cynthia E. Will
Affiliation: FaMAF and CIEM, Universidad Nacional de Córdoba, Córdoba, Argentina
Email: cwill@mate.uncor.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04757-0
PII: S 0002-9947(08)04757-0
Received by editor(s): March 22, 2007
Published electronically: November 25, 2008
Additional Notes: This research was supported by CONICET fellowships and grants from FONCyT and Fundación Antorchas.
Article copyright: © Copyright 2008 American Mathematical Society