From curves to currents

Geodesic currents are measures introduced by Bonahon in 1986 that realize a suitable closure of the space of closed curves on a surface. Bonahon proved that intersection number and hyperbolic length for curves extend to geodesic currents. Since then, many other functions defined on the space of curves have been extended to currents, such as negatively curved lengths, lengths from singular flat structures or stable lengths for surface groups. In this talk, we explain how a function defined on the space of curves satisfying some simple conditions can be extended continuously to geodesic currents. The most important of these conditions is that the function decreases under smoothing of essential crossings.
Our theorem subsumes previous extension results. Furthermore, it gives new extensions such as extremal length. This is joint work with Dylan Thurston.