Square-tiled cyclic covers

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 \cite{Forni:Matheus}) 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 found previously by Forni in genus three also corresponds to a square-tiled cyclic cover \cite{ForniSurvey}). We present several new examples of Teichmüller curves in strata of holomorphic and meromorphic quadratic differentials with maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichmüller curves with maximally degenerate spectrum. We prove that this is indeed the case within the class of square-tiled cyclic covers.

34 pages, 6 figures. Final version incorporating referees comments. In particular, a gap in the previous version was corrected. This file uses the journal's class file (jmd.cls), so that it is very similar to published version

References in corpus (3)

• Connected components of the moduli spaces of Abelian differentials with prescribed singularities
• Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials
• Euler characteristics of Teichmüller curves in genus two

Cited by in corpus (14)

• Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow
• A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces
• Zero Lyapunov exponents and monodromy of the Kontsevich-Zorich cocycle
• Lyapunov spectrum of invariant subbundles of the Hodge bundle
• Deviation of ergodic averages for substitution dynamical systems with eigenvalues of modulus one
• Eigenvalues of Curvature, Lyapunov exponents and Harder-Narasimhan filtrations
• Zero Lyapunov exponents of the Hodge bundle
• Loci in strata of meromorphic differentials with fully degenerate Lyapunov spectrum
• Weierstrass filtration on Teichmüller curves and Lyapunov exponents: Upper bounds
• On the Non-Uniform Hyperbolicity of the Kontsevich-Zorich Cocycle for Quadratic Differentials
• Some examples of isotropic SL(2,R)-invariant subbundles of the Hodge bundle
• Square-tiled tori
• On the construction of minimal foliations by hyperbolic surfaces on 3-manifolds
• The Extrinsic Primitive Torsion Problem