Advertisement

Trees and ultrametric Möbius structures

  • Jonas BeyrerEmail author
  • Victor Schroeder
Research Articles
  • 44 Downloads

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.

Key words

trees Möbius structures ultrametrics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. Bonk and O. Schramm, “Embeddings of Gromov hyperbolic spaces,” Geom. Func. Anal. 10, 266–306 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M. Bourdon, “Structure conforme au bord et flot géodésique d’un CAT(-1)-espace,” L’Einseignement Mathématique 41, 63–102 (1995).zbMATHGoogle Scholar
  5. 5.
    S. Buyalo and V. Schroeder, Elements of Asymptotic Geometry, EMS Monographs in Mathematics, 209 pages (2007).CrossRefzbMATHGoogle Scholar
  6. 6.
    I. Chiswell, Introduction to Λ-Trees (World Scientific, Singapore, 2001).CrossRefzbMATHGoogle Scholar
  7. 7.
    T. Foertsch and V. Schroeder, “Hyperbolicity, CAT(-1)-spaces and Ptolemy inequality,” Math. Ann. 350 (2), 339–356 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    B. Hughes, “Trees and ultrametric spaces: a categorical equivalence,” Advan. Math. 189, 148–191 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Z. Ibragimov, “Möbius maps between ultrametric spaces are local similarities,” Ann. Acad. Sci. Fenn.Math. 37, 309–317 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    J. Väisälä, “Gromov hyperbolic spaces,” Expo.Math. 23 (3), 187–311 (2005).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZürichSwitzerland

Personalised recommendations