Dual spaces of geodesic currents

Abstract Every geodesic current on a hyperbolic surface has an associated dual space. If the current is a lamination, this dual embeds isometrically into a real tree. We show that, in general, the dual space is a Gromov hyperbolic metric tree‐graded space, and express its Gromov hyperbolicity constant in terms of the geodesic current. In the case of geodesic currents with no atoms and full support, such as those coming from certain higher rank representations, we show the duals are homeomorphic to the surface. We also analyze the completeness of the dual and the properties of the action of the fundamental group of the surface on the dual. Furthermore, we compare two natural topologies in the space of duals.