Journal of Modern Dynamics (JMD)

Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces

Pages: 405 - 426, Issue 3, July 2012      doi:10.3934/jmd.2012.6.405

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Alex Wright - Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637, United States (email)

Abstract: We consider normal covers of $\mathbb{C}P^1$ with abelian deck group and branched over at most four points. Families of such covers yield arithmetic Teichmüller curves, whose period mapping may be described geometrically in terms of Schwarz triangle mappings. These Teichmüller curves are generated by abelian square-tiled surfaces.
    We compute all individual Lyapunov exponents for abelian square-tiled surfaces, and demonstrate a direct and transparent dependence on the geometry of the period mapping. For this we develop a result of independent interest, which, for certain rank two bundles, expresses Lyapunov exponents in terms of the period mapping. In the case of abelian square-tiled surfaces, the Lyapunov exponents are ratios of areas of hyperbolic triangles.

Keywords:  Teichmüller curve, Schwarz triangle mapping, Lyapunov exponent.
Mathematics Subject Classification:  Primary: 32G15; Secondary: 37D, 32G20, 30F30.

Received: May 2012;      Available Online: October 2012.