Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials
Anton Zorich
Journal of Modern Dynamics 2008, 2(1): 139-185 doi: 10.3934/jmd.2008.2.139
Moduli spaces of Abelian and quadratic differentials are stratified by multiplicities of zeroes; connected components of the strata correspond to ergodic components of the Teichmüller geodesic flow. It is known that the strata are not necessarily connected; the connected components were recently classified by M. Kontsevich and the author and by E. Lanneau. The strata can be also viewed as families of flat metrics with conical singularities and with $\mathbb Z$/$2 \mathbb Z$-holonomy.
    For every connected component of each stratum of Abelian and quadratic differentials we construct an explicit representative which is a Jenkins–Strebel differential with a single cylinder. By an elementary variation of this construction we represent almost every Abelian (quadratic) differential in the corresponding connected component of the stratum as a polygon with identified pairs of edges, where combinatorics of identifications is explicitly described.
    Specifically, the combinatorics is expressed in terms of a generalized permutation. For any component of any stratum of Abelian and quadratic differentials we construct a generalized permutation in the corresponding extended Rauzy class.
keywords: Teichmüller geodesic flow interval-exchange transformation Jenkins-Strebel differential moduli space of quadratic differentials Rauzy class. spin structure
Square-tiled cyclic covers
Giovanni Forni Carlos Matheus Anton Zorich
Journal of Modern Dynamics 2011, 5(2): 285-318 doi: 10.3934/jmd.2011.5.285
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: Kontsevich--Zorich cocycle Teichmüller geodesic flow square-tiled surfaces.
Lyapunov spectrum of square-tiled cyclic covers
Alex Eskin Maxim Kontsevich Anton Zorich
Journal of Modern Dynamics 2011, 5(2): 319-353 doi: 10.3934/jmd.2011.5.319
A cyclic cover over $CP^1$ branched at four points inherits a natural flat structure from the "pillow" flat structure on the basic sphere. We give an explicit formula for all individual Lyapunov exponents of the Hodge bundle over the corresponding arithmetic Teichmüller curve. The key technical element is evaluation of degrees of line subbundles of the Hodge bundle, corresponding to eigenspaces of the induced action of deck transformations.
keywords: Hodge norm moduli space of quadratic differentials Lyapunov exponent cyclic cover. Teichmüller geodesic flow

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