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

Abstract

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.

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Acknowledgments

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.

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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.

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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

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Keywords

  • Rank one symmetric spaces
  • Möbius structure
  • Ptolemy spaces

Mathematics Subject Classification

  • 53C35
  • 53C24