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

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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 \({\mathcal{H}^{\rm hyp}(4)}\).

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Authors and Affiliations

  1. IMB Bordeaux-Université de Bordeaux, 351, Cours de la Libération, 33405, Talence Cedex, France

    Duc-Manh Nguyen

  2. Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL, 60637, USA

    Alex Wright

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  1. Duc-Manh Nguyen
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  2. Alex Wright
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Correspondence to Duc-Manh Nguyen.

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Nguyen, DM., Wright, A. Non-Veech surfaces in \({\mathcal{H}^{\rm hyp}(4)}\) are generic. Geom. Funct. Anal. 24, 1316–1335 (2014). https://doi.org/10.1007/s00039-014-0297-0

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  • DOI: https://doi.org/10.1007/s00039-014-0297-0

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