Square-tiled cyclic covers

A cyclic cover of the complex projective line branched at four appropriatepoints has a natural structure of a square-tiled surface. We describethe combinatorics of such a square-tiled surface, the geometry of the correspondingTeichmüller curve, and compute the Lyapunov exponents of thedeterminant bundle over the Teichmüller curve with respect to the geodesicflow. This paper includes a new example (announced by G. Forni and C.Matheus in [17]) of a Teichmüller curve of a square-tiled cyclic cover in astratum of Abelian differentials in genus four with a maximally degenerateKontsevich-Zorich spectrum (the only known example in genus three foundpreviously by Forni also corresponds to a square-tiled cyclic cover [15]). Wepresent several new examples of Teichmüller curves in strata of holomorphicand meromorphic quadratic differentials with a maximally degenerateKontsevich-Zorich spectrum. Presumably, these examples cover all possibleTeichmüller curves with maximally degenerate spectra. We prove that this isindeed the case within the class of square-tiled cyclic covers.