Document Type
Article
Publication Date
11-30-2017
Publication Title
Israel Journal of Mathematics
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 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.
Volume
223
First Page
53
Last Page
74
DOI
10.1007/s11856-017-1619-3
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Rights
Licensed to Smith College and distributed CC-BY under the Smith College Faculty Open Access Policy.
Recommended Citation
Cahn, Patricia; Fanoni, Federica; and Petri, Bram, "Mapping Class Group Orbits of Curves with Self-Intersections" (2017). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/53
Comments
Peer reviewed accepted manuscript.