Skip to main content
SpringerLink
Log in

The geometry at infinity of a hyperbolic Riemann surface of infinite type

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

Abstract

We study geodesics on planar Riemann surfaces of infinite type having a single infinite end. Of particular interest is the class of geodesics that go out the infinite end in a most efficient manner. We investigate properties of these geodesics and relate them to the structure of the boundary of a Dirichlet polygon for a Fuchsian group representing 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. Ahlfors L.V. and Sario L. (1960). Riemann Surfaces. Princeton University Press, Princeton

    Google Scholar 

  2. Basmajian A. (1990). Constructing pairs of pants. Ann. Acad. Sci. Fen. Math. 15: 65–74

    MathSciNet  Google Scholar 

  3. Basmajian A. (1993). Hyperbolic structures for surfaces of infinite type. Trans. Amer. Math Soc. 336(1): 421–444

    Article  MathSciNet  Google Scholar 

  4. Beardon A.F. (1983). The Geometry of Discrete Groups. Springer-Verlag, Berlin

    Google Scholar 

  5. Haas A. (1996). Dirichlet points, Garnett points and Infinite ends of hyperbolic surfaces. I. Ann. Acad. Sci. Fenn. Math. 21(1): 3–29

    MathSciNet  Google Scholar 

  6. He Z.-X. and Schramm O. (1993). Fixed points, Koebe uniformization and circle packings. Ann. Math. 137(2): 369–406

    Article  MathSciNet  Google Scholar 

  7. Matelski J.P. (1976). A compactness theorem for Fuchsian groups of the second kind. Duke Math. J. 43(4): 829–840

    Article  MathSciNet  Google Scholar 

  8. Nicholls P.J. and Waterman P.L. (1990). The boundary of convex fundamental domains of Fuchsian groups. Ann. Acad. Sci. Fenn. Ser.A I Math. 15(1): 11–25

    MathSciNet  Google Scholar 

  9. Sullivan D. (1981). On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. In: Kra, I. and Maskit, B. (eds) Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Ann. of Math. Studies 97, pp 465–496. Princeton University Press, Princeton

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA

    Andrew Haas

  2. Department of Mathematics and Computer Science, Connecticut College, 270 Mohegan Ave, CT COLL Box 5596, New London, CT, 06320, USA

    Perry Susskind

Authors
  1. Andrew Haas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Perry Susskind
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Perry Susskind.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Haas, A., Susskind, P. The geometry at infinity of a hyperbolic Riemann surface of infinite type. Geom Dedicata 130, 1–24 (2007). https://doi.org/10.1007/s10711-007-9195-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-007-9195-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation