Bonahon introduced the space of geodesic currents, a space that contains all closed weighted closed curves as a dense subspace (generalizing measured laminations, which contain weighted simple closed curves). We give a simple criterion for when a function on weighted curves extends to a continuous function on geodesic currents; the key restriction is that it is monotone under smoothing a crossing. This puts all known results of functions extending to geodesic currents in a unified framework.

As a corollary of this result and work of Rafi-Souto, we can, for instance, give asymptotics for the number of curves of a given topological type with bounded extremal length. (For hyperbolic length, the corresponding result is a theorem of Mirzakhani.)

This is joint work with Didac Martinez-Granado.