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

Abstract

Mobius structure (on a set \(X\)) is a class of metrics having the same cross-ratios. A Mobius 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 Mobius 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 Mobius geometry: Let \(X\) be a compact Ptolemy space which contains a Ptolemy circle and allows many space inversions. Then \(X\) is Mobius equivalent to the boundary at infinity of a rank one symmetric space.