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Expansive dynamics and hyperbolic geometry

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Abstract

LetM be a compact Riemannian manifold with no conjugate points such that its geodesic flow is expansive. Then we show that the universal Riemannian covering ofM is a hyperbolic geodesic space according to the definition of M. Gromov. This allows us to extend a series of relevant geometric and topological properties of negatively curved manifolds toM and in particular, geometric group theory applies to the fundamental group ofM.

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References

  1. Anosov, D.: Geodesic flow on closed Riemannian manifolds of negative curvature. Trudy Math. Inst. Steklov 90. (1967).

  2. Ballman, W., Gromov, M., Schroeder, V.: Manifolds of non-positive curvature. Boston, Birkhauser 1985.

    Google Scholar 

  3. Ballman, W., Ghys, E. and others: Sur les groups hyperboliques d'après Mikhael Gromov. Preprint. Workshop on group theory from a geometrical viewpoint. I.C.T.P. 1990.

  4. Busemann, H.: The geometry of geodesics. Academic Press, 1955, New York.

    Google Scholar 

  5. Eberlein, P.: Geodesic flows in certain manifolds without conjugate points. Trans. of the Am. Math. Society. 167 (1972) 151–170.

    Google Scholar 

  6. Eberlein, P.: When is a goodesic flow of Anosov type I. Journal of Diff. Geom. 8 (1973) 437–463.

    Google Scholar 

  7. Gromoll, D., Klingenberg, W., Meyer, W.: Riemannsche Geometric in Grossen, Lecture Notes in Math. Vol. 55, Springer-Verlag, Heidelberg 1968.

    Google Scholar 

  8. Ghys, E., De La Harpe, P.: Sur les groupes hyperboliques d'après M. Gromov. Progress in Mathematics. Ghys, De La Harpe Editors 1991.

  9. Gromov, M.: Hyperbolic groups. Essays in group theory 8, 75-264 S.M. Gersten Editor. Springer-Verlag 1987.

  10. Pesin, Ja. B.: Geodesic flows on closed Riemannian manifolds without focal points. Math. USSR Izvestija, 11, 6 (1977).

    Google Scholar 

  11. Ruggiero, R.: Expansive geodesic flows in manifolds with no conjugate points. Preprint 1991.

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Authors and Affiliations

  1. Departamento de Matemática, Pontificia Universidade Católica do Rio de Janeiro, P.U.C., Rua Marquês de São Vicente, 225, 22453, Rio de Janeiro, Brasil

    Rafael O. Ruggiero

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  1. Rafael O. Ruggiero
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Ruggiero, R.O. Expansive dynamics and hyperbolic geometry. Bol. Soc. Bras. Mat 25, 139–172 (1994). https://doi.org/10.1007/BF01321305

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  • DOI: https://doi.org/10.1007/BF01321305

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