Didac Martinez Granado

Geodesic currents and the smoothing property

Geodesic currents are measures that realize a closure of the
space of curves on a closed surface.
Bonahon introduced geodesic currents in 1986, showed that geometric
intersection number extends to geodesic currents
and realized hyperbolic length of a curve as intersection number with a
geodesic current associated to the hyperbolic structure.
Since then, other functions on curves have been shown to extend to
geodesic currents. Some of them extend as intersection numbers, such as
negatively curved Riemannian lengths (Otal, 1990)
or word length w.r.t. simple generating sets of a surface group
(Erlandsson, 2016). Some other functions aren't intersection numbers but
extend continuously (Erlandsson-Parlier-Souto, 2016), such as word
length w.r.t. non-simple generating sets or extremal length of curves.
In this talk we present two results. The first is a criterion for a
function on curves to extend continuously to geodesic currents. The
second is a characterization of functions that arise as intersection
numbers. This is joint work with Dylan Thurston.