Square-tiled cyclic covers

Giovanni Forni, Carlos Matheus, Anton Zorich

Research output: Contribution to journalArticlepeer-review

28 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)285-318
Number of pages34
JournalJournal of Modern Dynamics
Volume5
Issue number2
DOIs
Publication statusPublished - Apr 2011
Externally publishedYes

Keywords

  • Kontsevich-zorich cocycle
  • Square-tiled surfaces
  • Teichmüller geodesic flow

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