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Geometriae Dedicata

, Volume 172, Issue 1, pp 1–45 | Cite as

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

  • Sergei BuyaloEmail author
  • Viktor Schroeder
Original Paper
  • 212 Downloads

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.

Keywords

Rank one symmetric spaces Möbius structure Ptolemy spaces 

Mathematics Subject Classification

53C35 53C24 

Notes

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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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.St. Petersburg Department of SteklovMathematical Institute RASSt. PetersburgRussia
  2. 2.Institut für MathematikUniversität ZürichZurichSwitzerland

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