Skip to main content
SpringerLink
Log in

Gromov hyperbolicity of Denjoy Domains

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

In this paper we characterize the Gromov hyperbolicity of the double of a metric space. This result allows to give a characterization of the hyperbolic Denjoy domains, in terms of the distance to \(\mathbb{R}\) of the points in some geodesics. In the particular case of trains (a kind of Riemann surfaces which includes the flute surfaces), we obtain more explicit criteria which depend just on the lengths of what we have called fundamental geodesics.

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. Ahlfors L., Sario L. (1960) Riemann Surfaces. Princeton Univ. Press, Princeton, New Jersey

    MATH  Google Scholar 

  2. Aikawa H. (2003) Positive harmonic functions of finite order in a Denjoy type domain. Proc. Am. Math. Soc. 131, 3873–3881

    Article  MATH  MathSciNet  Google Scholar 

  3. Alvarez V., Pestana D., Rodríguez J.M. (1999) Isoperimetric inequalities in Riemann surfaces of infinite type. Rev. Mat. Iberoamericana 15, 353–427

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  6. Balogh Z.M., Buckley S.M. (2003) Geometric characterizations of Gromov hyperbolicity. Invent. Math. 153, 261–301

    Article  MATH  MathSciNet  Google Scholar 

  7. Basmajian A. (1990) Constructing pair of pants. Ann. Acad. Sci. Fenn. Ser. AI 15, 65–74

    MATH  MathSciNet  Google Scholar 

  8. Basmajian A. (1993) Hyperbolic structures for surfaces of infinite type. Trans. Am. Math. Soc. 336, 421–444

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  10. Bishop R., O’Neill B. (1969) Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1–49

    Article  MATH  MathSciNet  Google Scholar 

  11. Bonk M. (1996) Quasi-geodesics segments and Gromov hyperbolic spaces. Geom. Dedicata 62, 281–298

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  13. Cantón A., Fernández J.L., Pestana D., Rodríguez J.M. (2001) On harmonic functions on trees. Poten. Anal. 15, 199–244

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  15. Coornaert M., Delzant T., Papadopoulos A. (1989) Notes sur les groups hyperboliques de Gromov. I.R.M.A., Strasbourg

    Google Scholar 

  16. Eberlein, P.: Surfaces of nonpositive curvature. Mem. Am. Math. Soc. 20(218), (1979)

  17. Fenchel W. (1989) Elementary Geometry in Hyperbolic Space. Walter de Gruyter, Berlin-New York

    MATH  Google Scholar 

  18. Fernández J.L., Rodríguez J.M. (1992) Area growth and Green’s function of Riemann surfaces. Arkiv för matematik 30, 83–92

    Article  MATH  Google Scholar 

  19. Garnett J., Jones P. (1985) The Corona theorem for Denjoy domains. Acta Math. 155, 27–40

    Article  MATH  MathSciNet  Google Scholar 

  20. Ghys E., de la Harpe P.: Sur les Groupes Hyperboliques d’après Mikhael Gromov. Progress in Mathematics, Vol. 83. Birkhäuser (1990)

  21. González M.J. (1998) An estimate on the distortion of the logarithmic capacity. Proc. Am. Math. Soc. 126, 1429–1431

    Article  MATH  Google Scholar 

  22. Haas A. (1996) Dirichlet points, Garnett points, and infinite ends of hyperbolic surfaces I. Ann. Acad. Sci. Fenn. Ser. AI 21, 3–29

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  26. Portilla A., Rodríguez J.M., Tourís E. (2004) Gromov hyperbolicity through decomposition of metric spaces II. J. Geom. Anal. 14, 123–149

    MATH  MathSciNet  Google Scholar 

  27. Portilla A., Rodríguez J.M., Tourís E. (2004) The topology of balls and Gromov hyperbolicity of Riemann surfaces. Diff. Geom. Appl. 21, 317–335

    Article  MATH  Google Scholar 

  28. Portilla A., Rodríguez J.M., Tourís E. (2006) The role of funnels and punctures in the Gromov hyperbolicity of Riemann surfaces. Proc. Edinburgh Math. Soc. 49, 399–425

    Article  MATH  Google Scholar 

  29. Soardi P.M. (1993) Rough isometries and Dirichlet finite harmonic functions on graphs. Proc. Am. Math. Soc. 119, 1239–1248

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911, Leganés, Madrid, Spain

    Ana Portilla, Jose M. Rodriguez & Eva Touris

  2. Departamento de Análisis Matemático, Facultad de Ciencias, Campus de Teatinos, 29071, Málaga, Spain

    Venancio Alvarez

Authors
  1. Venancio Alvarez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ana Portilla
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jose M. Rodriguez
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eva Touris
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jose M. Rodriguez.

Additional information

Research partially supported by three grants from M.E.C. (MTM 2006-11976, MTM 2006-13000-C03-02 and MTM 2004-21420-E), Spain.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alvarez, V., Portilla, A., Rodriguez, J.M. et al. Gromov hyperbolicity of Denjoy Domains. Geom Dedicata 121, 221–245 (2006). https://doi.org/10.1007/s10711-006-9102-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-006-9102-z

Keywords

Mathematical Subject Classifications (2000)

Search

Navigation