Journal of Modern Dynamics (JMD)

Square-tiled cyclic covers

Pages: 285 - 318, Issue 2, April 2011      doi:10.3934/jmd.2011.5.285

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Giovanni Forni - Department of Mathematics, University of Maryland, College Park, MD 20742-4015, United States (email)
Carlos Matheus - Collège de France, 3 Rue d’Ulm, Paris, CEDEX 05, France (email)
Anton Zorich - IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France (email)

Abstract: A cyclic cover of the complex projective line branched at four appropriate 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. 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 a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example in genus three found previously by Forni also corresponds to a square-tiled cyclic cover [15]. We present several new examples of Teichmüller curves in strata of holomorphic and meromorphic quadratic differentials with a maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichmüller curves with maximally degenerate spectra. We prove that this is indeed the case within the class of square-tiled cyclic covers.

Keywords:  Teichmüller geodesic flow, Kontsevich--Zorich cocycle, square-tiled surfaces.
Mathematics Subject Classification:  Primary: 37D25, 37D40; Secondary: 14F05.

Received: July 2010;      Revised: June 2011;      Available Online: July 2011.