Mapping class group orbits of curves with self-intersections

TL;DR

This paper analyzes the asymptotic behavior of mapping class group orbits of curves with self-intersections on surfaces, revealing growth rates and relationships between homotopy and isotopy classes as surface complexity increases.

Contribution

It provides new asymptotic formulas for counting orbits of curves with self-intersections and minimal genus, and shows most homotopic curves are also isotopic.

Findings

Asymptotic count of orbits with bounded self-intersections

Most homotopic curves are isotopic

Polynomial bounds on orbits containing short curves

Abstract

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 a 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.