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 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.