### Geodesic currents and the smoothing property

Geometry and Topology Seminar

15th February 2021, 4:00 pm – 5:00 pm

Zoom seminar, if interested, please email one of the organisers to gain access to the Zoom link

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.

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