Israel Journal of Mathematics
We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity.
We also consider the minimal genus of a subsurface that contains the curve. We determine the asymptotic number of orbits of curves with a fixed minimal genus and a bounded self-intersection number, as the complexity of the surface tends to infinity.
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 given hyperbolic surface that can contain short curves. For a fixed length, this bound is polynomial in the signature of the surface.
The arguments we use are based on counting embeddings of ribbon graphs.
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This work is licensed under a Creative Commons Attribution 4.0 International License.
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Cahn, Patricia; Fanoni, Federica; and Petri, Bram, "Mapping Class Group Orbits of Curves with Self-Intersections" (2017). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.