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THE BOUW-MÖLLER LATTICE SURFACES AND EIGENVECTORS OF GRID GRAPHS

By W. Patrick Hooper

Abstract

Abstract. This article investigates a family of translation surfaces whose Veech groups are lattices closely related to triangle groups. Many of these examples were discovered by Bouw and Möller through non-elementary means, and lack a simple elementary description. This article describes many of the Bouw-Möller examples by gluing together polygons in simple ways, and discovers some closely related new examples. We connect these examples to the principal eigenvectors of grid graphs, and provide analogous constructions of infinite genus translation surfaces with the lattice property. A Teichmüller curve is a totally geodesic embedding of a complete two-dimensional hyperbolic orbifold into the moduli space of surfaces genus g equipped with the Teichmüller metric. Veech found the first examples of these objects [Vee89], and found Teichmüller curves in each genus g ≥ 2. Since then, there has been interest in finding more examples and classifying these objects. (See [KS00], [McM03], [Cal04] and [McM06] for instance.) Teichmüller curves are naturally associated to flat structures and translation surfaces with the lattice property. See section 1.1. Relatively recently, Bouw and Möller found Teichmüller curves isometric to H 2 /Γ for Γ

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