Journal of Modern Dynamics (JMD)

Shimura and Teichmüller curves

Pages: 1 - 32, Issue 1, January 2011      doi:10.3934/jmd.2011.5.1

       Abstract        References        Full Text (645.9K)       Related Articles

Martin Möller - Institut für Mathematik, Johann Wolfgang Goethe-Universität Frankfurt, Robert-Mayer-Str. 6–8, 60325 Frankfurt am Main, Germany (email)

Abstract: We classify curves in the moduli space of curves $M_g$ that are both Shimura and Teichmüller curves: for both $g=3$ and $g=4$ there exists precisely one such curve, for $g=2$ and $g \geq 6$ there are no such curves.
   We start with a Hodge-theoretic description of Shimura curves and of Teichmüller curves that reveals similarities and differences of the two classes of curves. The proof of the classification relies on the geometry of square-tiled coverings and on estimating the numerical invariants of these particular fibered surfaces.
   Finally, we translate our main result into a classification of Teichmüller curves with totally degenerate Lyapunov spectrum.

Keywords:  Shimura variety, square-tiled surface, Lyapunov spectrum.
Mathematics Subject Classification:  Primary: 14H15; Secondary: 14G35.

Received: July 2009;      Revised: January 2011;      Available Online: April 2011.