Mapping class group orbits of curves with self-intersections

Patricia Cahn, Federica Fanoni, Bram Petri

Published 2016-03-02Version 1

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of number of such orbits, on a genus g surface with k self-intersections, for k bounded and $g\to\infty$. To do this we count embeddings of ribbon graphs. As a corollary of our methods, we obtain that most curves that are homotopic are also isotopic. Furthermore, using a theorem by Basmajian, we get a bound on the number of mapping class group orbits on a hyperbolic surface that can contain short curves. For a fixed length, this bound is polynomial in the genus of the surface.