Möbius characterization of the boundary at infinity of rank one symmetric spaces


Möbius structure (on a set \(X\)) is a class of metrics having the same cross-ratios. A Möbius structure is Ptolemaic if it is invariant under inversion operations. The boundary at infinity of a \(\mathrm{CAT }(-1)\) space is in a natural way a Möbius space, which is Ptolemaic. We give a free of classification proof of the following result that characterizes the rank one symmetric spaces of noncompact type purely in terms of their Möbius geometry: Let \(X\) be a compact Ptolemy space which contains a Ptolemy circle and allows many space inversions. Then \(X\) is Möbius equivalent to the boundary at infinity of a rank one symmetric space.

This is a preview of subscription content, access via your institution.


  1. 1.

    Buckley, S., Falk, K., Wraith, D.: Ptolemaic spaces and CAT(0). Glasg. Math. J. 51(2), 301–314 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Cowling, M., Dooley, A., Korányi, A., Ricci, F.: An approach to symmetric spaces of rank one via groups of Heisenberg type. J. Geom. Anal. 8, 199–237 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Foertsch, T., Lytchak, A., Schroeder, V.: Nonpositive curvature and the Ptolemy inequality. Int. Math. Res. Not. IMRN 22, 15 (2007). Art. ID rnm100

    Google Scholar 

  4. 4.

    Foertsch, T., Schroeder, V.: Hyperbolicity, \(\text{ CAT }(-1)\)-spaces and Ptolemy inequality. Math. Ann. 350(2), 339–356 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Foertsch, T., Schroeder, V.: A Möbius Characterization of Metric Spheres, arXiv:math/1008.3250 (2010), to appear in Manuscripta Math

  6. 6.

    Heintze, E.: On homogeneous manifolds of negative curvature. Math. Ann. 211, 23–34 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  8. 8.

    Kay, D.: Ptolemaic metric spaces and the characterization of geodesics by vanishing metric curvature, Ph.D. thesis, Michigan State Univ., East Lansing, MI (1963)

  9. 9.

    Kramer, L.: Two-transitive Lie groups. J. Reine Angew. Math. 563, 83–113 (2003)

    MATH  MathSciNet  Google Scholar 

  10. 10.

    Schoenberg, I.: A remark on M. M. Day’s characterization of inner-product spaces and a conjecture of L. M. Blumenthal. Proc. Am. Math. Soc. 3, 961–964 (1952)

    MATH  MathSciNet  Google Scholar 

  11. 11.

    Siebert, E.: Contractive automorphisms on locally compact groups. Mathematische Zeitschrift 191, 73–90 (1986)

    Article  MATH  MathSciNet  Google Scholar 

Download references


We are thankful to L. Kramer for informing us about 2-transitive group actions. The first author is also grateful to the University of Zürich for hospitality and support. We thank the referee for his or her very helpful comments and suggestions.

Author information



Corresponding author

Correspondence to Sergei Buyalo.

Additional information

Sergei Buyalo: Supported by RFBR Grant 11-01-00302-a and SNF Grant 20-119907/1.

Viktor Schroeder: Supported by Swiss National Science Foundation Grant 20-119907/1.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Buyalo, S., Schroeder, V. Möbius characterization of the boundary at infinity of rank one symmetric spaces. Geom Dedicata 172, 1–45 (2014). https://doi.org/10.1007/s10711-013-9906-6

Download citation


  • Rank one symmetric spaces
  • Möbius structure
  • Ptolemy spaces

Mathematics Subject Classification

  • 53C35
  • 53C24