Abstract
We study the topology and geometry of the moduli space of immersions of ellipses into a translation surface. The frontier of this space is naturally stratified by the number of ‘cone points’ that an ellipse meets. The stratum consisting of ellipses that meet three cone points is naturally a two dimensional (non-manifold) polygonal cell complex. We show that the topology of this cell-complex together with the eccentricity and direction of each of its vertices determines the translation surface up to homothety. As a corollary we characterize the Veech group of the translation surface in terms of automorphisms of this polygonal cell complex.
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S. A. Broughton thanks Indiana University for its hospitality. C. Judge thanks the Max-Planck Institut für Mathematik (Bonn), the Institut Fourier, and the École Polytechnique Fédéral de Lausanne for their hospitality and support.
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Broughton, S.A., Judge, C. Ellipses in translation surfaces. Geom Dedicata 157, 111–151 (2012). https://doi.org/10.1007/s10711-011-9602-3
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DOI: https://doi.org/10.1007/s10711-011-9602-3