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A special class of nilmanifolds admitting an Anosov diffeomorphism
Author(s):
Karel
Dekimpe;
Wim
Malfait
Journal:
Proc. Amer. Math. Soc.
128
(2000),
2171-2179.
MSC (1991):
Primary 58F15, 57R50, 20F34, 20F18
Posted:
November 23, 1999
MathSciNet review:
1664349
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Abstract:
A nilmanifold admits an Anosov diffeomorphism if and only if its fundamental group (which is finitely generated, torsion-free and nilpotent) supports an automorphism having no eigenvalues of absolute value one. Here we concentrate on nilpotency class 2 and fundamental groups whose commutator subgroup is of maximal torsion-free rank. We prove that the corresponding nilmanifold admits an Anosov diffeomorphism if and only if the torsion-free rank of the abelianization of its fundamental group is greater than or equal to 3.
References:
- 1.
- Anosov, D. Geodesic flow on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst., 90, 1969. MR 36:7157
- 2.
- Malfait, W. Anosov diffeomorphisms on nilmanifolds of dimension at most six. 1998. Preprint.
- 3.
- Manning, A. There are no new Anosov diffeomorphisms on tori. Amer. J. Math., 1974, 96 (3), pp. 1-99. MR 50:11324
- 4.
- Porteous, H. L. Anosov diffeomorphisms of flat manifolds. Topology, 1972, 11, pp. 307-315. MR 45:6035
- 5.
- Smale, S. Differentiable dynamical systems. Bull. Amer. Math. Soc., 1967, 73, pp. 747-817. MR 37:3598
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Additional Information:
Karel
Dekimpe
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven Campus Krotrijk, Universitaire Campus, B-8500 Kortrijk, Belgium
Email:
Karel.Dekimpe@kulak.ac.be
Wim
Malfait
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven Campus Krotrijk, Universitaire Campus, B-8500 Kortrijk, Belgium
Email:
Wim.Malfait@kulak.ac.be
DOI:
10.1090/S0002-9939-99-05337-X
PII:
S 0002-9939(99)05337-X
Keywords:
Anosov diffeomorphism,
nilmanifold,
free 2-step nilpotent Lie algebra,
hyperbolic automorphism
Received by editor(s):
August 5, 1998
Posted:
November 23, 1999
Additional Notes:
Both authors are Postdoctoral Fellows of the Fund for Scientific Research -- Flanders (Belgium) (F.W.O.).
Communicated by:
Józef Dodziuk
Copyright of article:
Copyright
2000,
American Mathematical Society
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