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Rigidity of the nonseparating and outer curve graph

  • Jesús Hernández Hernández
Original Article
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Abstract

Let \(S_{1}\) and \(S_{2}\) be orientable surfaces of finite topological type with empty boundary, both of genus at least 1 and \(n_{1}, n_{2} \ge 0\) punctures. We define the nonseparating and outer curve graph \(\mathcal {NO}(S_{i})\): a particular induced subgraph of the curve graph, which has the same large-scale geometry. We prove (under certain conditions on the complexity of \(S_{1}\) and \(S_{2}\)) that if \(\varphi : \mathcal {NO}(S_{1}) \rightarrow \mathcal {NO}(S_{2})\) is a superinjective map, then \(S_{1}\) is homeomorphic to \(S_{2}\) and \(\varphi \) is induced by a homeomorphism.

Keywords

Curve graph Rigidity Mapping class group Superinjective map 

Mathematics Subject Classification

20F65 57M07 

Notes

Acknowledgements

The author thanks his Ph.D. advisors, Javier Aramayona and Hamish Short, for their very helpful suggestions, talks, corrections, and specially for their patience while giving shape to this work. He also thanks Ferrán Valdez as well for a careful reading of this work. The author also thanks the referees for their very helpful suggestions and comments, which made this work more streamlined and readable.

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Copyright information

© Sociedad Matemática Mexicana 2018

Authors and Affiliations

  1. 1.Centro de Ciencias MatemáticasUNAM Universidad Nacional Autónoma de MéxicoMoreliaMexico

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