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Geometric and Functional Analysis

, Volume 24, Issue 4, pp 1316–1335 | Cite as

Non-Veech surfaces in \({\mathcal{H}^{\rm hyp}(4)}\) are generic

  • Duc-Manh NguyenEmail author
  • Alex Wright
Article

Abstract

We show that every surface in the component \({\mathcal{H}^{\rm hyp}(4)}\), that is the moduli space of pairs \({(M,\omega)}\) where M is a genus three hyperelliptic Riemann surface and \({\omega}\) is an Abelian differential having a single zero on M, is either a Veech surface or a generic surface, i.e. its \({{\rm GL}^{+}(2,\mathbb{R})}\)-orbit is either a closed or a dense subset of \({\mathcal{H}^{\rm hyp}(4)}\). The proof develops new techniques applicable in general to the problem of classifying orbit closures, especially in low genus. Combined with work of Matheus and the second author, a corollary is that there are at most finitely many non-arithmetic Teichmüller curves (closed orbits of surfaces not covering the torus) in \({\mathcal{H}^{\rm hyp}(4)}\).

Keywords

Modulus Space Orbit Closure Horizontal Cylinder Translation Surface Saddle Connection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.IMB Bordeaux-Université de BordeauxTalence CedexFrance
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA

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