Mapping class group orbits of curves with self-intersections
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...
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| Main Authors: | , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
February 2018
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| In: |
Israel journal of mathematics
Year: 2018, Volume: 223, Issue: 1, Pages: 53-74 |
| ISSN: | 1565-8511 |
| DOI: | 10.1007/s11856-017-1619-3 |
| Online Access: | Verlag, Volltext: http://dx.doi.org/10.1007/s11856-017-1619-3 Verlag, Volltext: https://link.springer.com/article/10.1007/s11856-017-1619-3 
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| Author Notes: | by Patricia Cahn and Federica Fanoni and Bram Petri |
| Summary: | 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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| Item Description: | First online: 30 November 2017 Gesehen am 16.03.2018 |
| Physical Description: | Online Resource |
| ISSN: | 1565-8511 |
| DOI: | 10.1007/s11856-017-1619-3 |