Geodesic currents and counting curves
Analysis and Geometry Seminar
24th October 2017, 3:00 pm – 4:00 pm
Howard House, 2nd Floor Seminar Room
A famous result by Mirzakhani gives the asymptotic growth of the number of curves on a hyperbolic surface of bounded length L, as L grows. If S is a surface of genus g equipped with a hyperbolic structure, she showed that the number of such curves on S (in each mapping class group orbit) is asymptotic to a constant times L6g-6. In this talk I will explain, through the use of geodesic currents, why the same asymptotics hold for other notions of length. In particular, if one measures length with respect to any Riemannian metric on S or the word metric with respect to any finite generating set of its fundamental group.