Moebius characterization of the boundary at infinity of rank one   symmetric spaces
  A Moebius structure (on a set X) is a class of metrics having the same cross-ratios. A Moebius structure is ptolemaic if it is invariant under inversion operations. The boundary at infinity of a CAT(-1) space is in a natural way a Moebius 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 Moebius geometry: Let X be a compact Ptolemy space which contains a Ptolemy circle and allows many space inversions. Then X is Moebius equivalent to the boundary at infinity of a rank one symmetric space. 
Sergei Buyalo, Viktor Schroeder
math.MG math.DG