Square-tiled cyclic covers
Résumé
A cyclic cover of the projective plane branched at four points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichmüller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichmüller curve with respect to the geodesic flow. We find a new example of a Teichmüller curve with a completely degenerate Lyapunov spectrum (the only known example found previously by G. Forni also corresponds to a cyclic cover). Presumably, these two examples cover all possible Teichmüller curves with completely degenerate Lyapunov spectrum.