Topological Characterizations of Geometrically Infinite Convergence Group Actions and Orbit Uniform Metrics

Two characterizations of geometric finiteness in terms of weaker conditions are known for actions on Gromov hyperbolic spaces. In this paper, we establish analogous characterizations for general convergence group actions. To this end, we introduce the notion of an orbit uniform metric. We prove that a point is either conical or bounded parabolic if and only if its orbit is discrete with respect to an orbit uniform metric. As a consequence, we characterize geometric infiniteness in terms of the uncountability of the set of non-conical limit points. We further characterize geometric infiniteness by the existence of escaping sequences of hyperbolic elements, thereby extending the corresponding result for actions on Gromov hyperbolic spaces.