Trees and ultrametric Möbius structures


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.

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  1. 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. 2.

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

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

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

    MathSciNet  Article  MATH  Google Scholar 

  4. 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. 5.

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

    Book  MATH  Google Scholar 

  6. 6.

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

    Book  MATH  Google Scholar 

  7. 7.

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

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

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

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

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

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

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

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Jonas Beyrer.

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The text was submitted by the authors in English.

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Beyrer, J., Schroeder, V. Trees and ultrametric Möbius structures. P-Adic Num Ultrametr Anal Appl 9, 247–256 (2017).

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Key words

  • trees
  • Möbius structures
  • ultrametrics