A Ferrand-Obata theorem for rank one parabolic geometries

The aim of this article is the proof of the following result: Let M be a connected manifold endowed with a regular Cartan geometry modelled on the boundary X of the d-dimensional real (resp. complex, resp. quaternionic, resp. octonionic) hyperbolic space. If the group of automorphisms of M does not act properly on M, then M is geometrically isomorphic to: - X if M is compact. - X minus a point in the other cases.