Shimura and Teichmüller curves

Abstract

We classify curves in the moduli space of curves $M_g$ that are both Shimuraand Teichmüller curves: for both $g=3$ and $g=4$ there exists precisely onesuch curve, for $g=2$ and $g \geq 6$ there are no such curves.   We start with a Hodge-theoretic description of Shimura curves and ofTeichmüller curves that reveals similarities and differences of the twoclasses of curves. The proof of the classification relies on the geometry ofsquare-tiled coverings and on estimating the numerical invariants of theseparticular fibered surfaces.   Finally, we translate our main result into a classification of Teichmüllercurves with totally degenerate Lyapunov spectrum.