Skip to main content
SpringerLink
Log in

Gromov hyperbolicity through decomposition of metric spaces

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We study the hyperbolicity of metric spaces in the Gromov sense. We deduce the hyperbolicity of a space from the hyperbolicity of its “building block components”. These results are valuable since they simplify notably the topology of the space and allow to obtain global results from local information. We also study how the punctures and the decomposition of a Riemann surface in Y-pieces and funnels affect the hyperbolicity of the surface.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. V. Ahlfors, Conformal Invariants, McGraw-Hill (New York, 1973).

    MATH  Google Scholar 

  2. V. Alvarez, D. Pestana and J. M. Rodríguez, Isoperimetric inequalities in Riemann surfaces of infinite type, Rev. Mat. Iber., 15 (1999), 353-427.

    MATH  Google Scholar 

  3. V. Alvarez and J. M. Rodríguez, Structure theorems for Riemann and topological surfaces, J. London Math. Soc., to appear.

  4. V. Alvarez, J. M. Rodríguez and V. A. Yakubovich, Subadditivity of p-harmonic “measure” on graphs, Michigan Math. J., 49 (2001), 47-64.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. W. Anderson, Hyperbolic Geometry, Springer (London, 1999).

    MATH  Google Scholar 

  6. Z. M. Balogh and S. M. Buckley, Geometric characterizations of Gromov hyperbolicity, Invent. Math., to appear.

  7. A. F. Beardon, The Geometry of Discrete Groups, Springer-Verlag (New York, 1983).

    MATH  Google Scholar 

  8. L. Bers, An inequality for Riemann surfaces, in: Differential Geometry and Complex Analysis, H. E. Rauch Memorial Volume, Springer-Verlag (New York, 1985).

    Google Scholar 

  9. M. Bonk, J. Heinonen and P. Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque, 270 (Paris, 2001).

  10. P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Birkhäuser (Boston, 1992).

    MATH  Google Scholar 

  11. A. Cantón, J. L. Fernández, D. Pestana and J. M. Rodríguez, On harmonic functions on trees, Potential Analysis, 15 (2001), 199-244.

    Article  MATH  MathSciNet  Google Scholar 

  12. I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press (New York, 1984).

    MATH  Google Scholar 

  13. J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in: Problems in Analysis, Princeton University Press (Princeton, 1970), pp. 195-199.

    Google Scholar 

  14. J. L. Fernández and M. V. Melián, Escaping geodesics of Riemannian surfaces, Acta Mathematica, 187 (2001), 213-236.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. L. Fernández and J. M. Rodríguez, The exponent of convergence of Riemann surfaces. Bass Riemann surfaces, Ann. Acad. Sci. Fenn. Series AI, 15 (1990), 165-183.

    MATH  Google Scholar 

  16. J. L. Fernández and J. M. Rodríguez, Area growth and Green's function of Riemann surfaces, Arkiv för Matematik, 30 (1992), 83-92.

    Article  MATH  Google Scholar 

  17. E. Ghys and P. de la Harpe, Sur les groupes hyperboliques d'aprìs Mikhael Gromov, in: Progress in Mathematics, Volume 83, Birkhäuser (Boston, 1990).

    Google Scholar 

  18. M. Gromov, Hyperbolic groups, Essays in group theory, Edited by S. M. Gersten, M.S.R.I. Publ. 8, Springer (New York, 1987), 75-263.

    Google Scholar 

  19. M. Gromov (with appendices by M. Katz, P. Pansu and S. Semmes), Metric Structures for Riemannian and Non-Riemannnian Spaces, Progress in Mathematics, vol. 152, Birkhäuser (Boston, 1999).

    Google Scholar 

  20. I. Holopainen and P. M. Soardi, p-harmonic functions on graphs and manifolds, Manuscripta Math., 94 (1997), 95-110.

    MATH  MathSciNet  Google Scholar 

  21. M. Kanai, Rough isometries and combinatorial approximations of geometries of non-compact Riemannian manifolds, J. Math. Soc. Japan, 37 (1985), 391-413.

    MATH  MathSciNet  Google Scholar 

  22. M. Kanai, Rough isometries and the parabolicity of Riemannian manifolds, J. Math. Soc. Japan, 38 (1986), 227-238.

    MATH  MathSciNet  Google Scholar 

  23. M. Kanai, Analytic inequalities and rough isometries between non-compact Riemannian manifolds, Curvature and Topology of Riemannian manifolds (Katata, 1985), Lecture Notes in Math. 1201, Springer (New York, 1986), 122-137.

    Google Scholar 

  24. I. Kra, Automorphic Forms and Kleinian Groups, W. A. Benjamin (1972).

  25. P. J. Nicholls, The Ergodic Theory of Discrete Groups, Lecture Notes Series 143, Cambridge University Press (Cambridge, 1989).

    MATH  Google Scholar 

  26. F. Paulin, On the critical exponent of a discrete group of hyperbolic isometries, Diff. Geom. Appl., 7 (1997), 231-236.

    Article  MATH  MathSciNet  Google Scholar 

  27. A. Portilla, J. M. Rodríguez and E. Tourís, Gromov hyperbolicity through decomposition of metric spaces II, J. Geom. Anal., to appear.

  28. A. Portilla, J. M. Rodríguez and E. Tourís, The topology of balls and Gromov hyperbolicity of Riemann surfaces, Diff. Geom. Appl., to appear.

  29. B. Randol, Cylinders in Riemann surfaces, Comment. Math. Helv., 54 (1979), 1-5.

    MATH  MathSciNet  Google Scholar 

  30. J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer-Verlag (New York, 1994).

    MATH  Google Scholar 

  31. J. M. Rodríguez, Isoperimetric inequalities and Dirichlet functions of Riemann surfaces, Publications Matemàtiques, 38 (1994), 243-253.

    MATH  Google Scholar 

  32. J. M. Rodríguez, Two remarks on Riemann surfaces, Publications Matemàtiques, 38 (1994), 463-477.

    MATH  Google Scholar 

  33. J. M. Rodríguez and E. Tourís, Simple closed geodesics determine the Gromov hyperbolicity of Riemann surfaces, to appear.

  34. J. M. Rodríguez and E. Tourís, Gromov hyperbolicity of Riemann surfaces, to appear.

  35. H. Shimizu, On discontinuous groups operating on the product of upper half-planes, Ann. of Math., 77 (1963), 33-71.

    Article  MATH  MathSciNet  Google Scholar 

  36. P. M. Soardi, Rough isometries and Dirichlet finite harmonic functions on graphs, Proc. Amer. Math. Soc., 119 (1993), 1239-1248.

    Article  MATH  MathSciNet  Google Scholar 

  37. D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Diff. Geom., 25 (1987), 327-351.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matematicas, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911, Leganes, Madrid, Spain

    José M. Rodríguez & Eva Tourís

Authors
  1. José M. Rodríguez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eva Tourís
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rodríguez, J.M., Tourís, E. Gromov hyperbolicity through decomposition of metric spaces. Acta Mathematica Hungarica 103, 107–138 (2004). https://doi.org/10.1023/B:AMHU.0000028240.16521.9d

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:AMHU.0000028240.16521.9d

Search

Navigation