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Shimura and Teichmüller curves

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  • We classify curves in the moduli space of curves $M_g$ that are both Shimura and Teichmüller curves: for both $g=3$ and $g=4$ there exists precisely one such curve, for $g=2$ and $g \geq 6$ there are no such curves.
       We start with a Hodge-theoretic description of Shimura curves and of Teichmüller curves that reveals similarities and differences of the two classes of curves. The proof of the classification relies on the geometry of square-tiled coverings and on estimating the numerical invariants of these particular fibered surfaces.
       Finally, we translate our main result into a classification of Teichmüller curves with totally degenerate Lyapunov spectrum.
    Mathematics Subject Classification: Primary: 14H15; Secondary: 14G35.


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