Skip to main content
SpringerLink
Log in

Trees and ultrametric Möbius structures

  • Research Articles
  • Published:
p-Adic Numbers, Ultrametric Analysis and Applications Aims and scope Submit manuscript

Abstract

We define the concept of an ultrametric Möbius space (Z,M) and show that the boundary at infinity of a nonelementary geodesically complete tree is naturally an ultrametric Möbius space. In addition, we construct to a given ultrametric Möbius space (Z,M) a nonelementary geodesically complete tree, unique up to isometry, with (Z,M) being its boundary at infinity. This yields a one-to-one correspondence.

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. M. Bestvina, “R-trees in topology, geometry and group theory,” in Handbook of Geometric Topology, pp. 55–91, edited by R. J. Daverman and R. B. Sher (North-Holland, Amsterdam, 2002).

    Google Scholar 

  2. K. Biswas, “On Möbius and conformal maps between boundaries of CAT(-1) spaces,” Annales de la Institut Fourier 65 (3), 1387–1422 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Bonk and O. Schramm, “Embeddings of Gromov hyperbolic spaces,” Geom. Func. Anal. 10, 266–306 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Bourdon, “Structure conforme au bord et flot géodésique d’un CAT(-1)-espace,” L’Einseignement Mathématique 41, 63–102 (1995).

    MATH  Google Scholar 

  5. S. Buyalo and V. Schroeder, Elements of Asymptotic Geometry, EMS Monographs in Mathematics, 209 pages (2007).

    Book  MATH  Google Scholar 

  6. I. Chiswell, Introduction to Λ-Trees (World Scientific, Singapore, 2001).

    Book  MATH  Google Scholar 

  7. T. Foertsch and V. Schroeder, “Hyperbolicity, CAT(-1)-spaces and Ptolemy inequality,” Math. Ann. 350 (2), 339–356 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Hughes, “Trees and ultrametric spaces: a categorical equivalence,” Advan. Math. 189, 148–191 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  9. Z. Ibragimov, “Möbius maps between ultrametric spaces are local similarities,” Ann. Acad. Sci. Fenn.Math. 37, 309–317 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Väisälä, “Gromov hyperbolic spaces,” Expo.Math. 23 (3), 187–311 (2005).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Zürich, Winterthurer Strasse 190, CH-8057, Zürich, Switzerland

    Jonas Beyrer & Victor Schroeder

Authors
  1. Jonas Beyrer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Victor Schroeder
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonas Beyrer.

Additional information

The text was submitted by the authors in English.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beyrer, J., Schroeder, V. Trees and ultrametric Möbius structures. P-Adic Num Ultrametr Anal Appl 9, 247–256 (2017). https://doi.org/10.1134/S207004661704001X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S207004661704001X

Key words

Search

Navigation