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

TL;DR

This paper characterizes the boundary at infinity of rank one symmetric spaces using Moebius geometry, showing that certain Ptolemy spaces with specific properties are equivalent to these boundaries.

Contribution

It provides a classification proof that identifies the boundary at infinity of rank one symmetric spaces purely through Moebius geometric properties.

Findings

Ptolemy spaces with Ptolemy circles and many inversions are equivalent to boundaries of rank one symmetric spaces

The boundary at infinity of a CAT(-1) space naturally forms a Moebius space that is ptolemaic

The paper offers a classification of these boundaries without relying on prior geometric assumptions

Abstract

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.