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Journal of Modern Dynamics (JMD)
 

The equation of the Kenyon-Smillie (2,3,4)-Teichmüller curve

Pages: 17 - 41, Volume 11, 2017      doi:10.3934/jmd.2017002

 
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Matteo Costantini - Institut für Mathematik, Goethe-Universität Frankfurt/Main, Robert-Mayer-Str. 6–8, 60325 Frankfurt, Germany (email)
André Kappes - Institut für Mathematik, Goethe-Universität Frankfurt/Main, Robert-Mayer-Str. 6–8, 60325 Frankfurt, Germany (email)

Abstract: We compute the algebraic equation of the universal family over the Kenyon-Smillie $(2,3,4)$-Teichmüller curve, and we prove that the equation is correct in two different ways. Firstly, we prove it in a constructive way via linear conditions imposed by three special points of the Teichmüller curve. Secondly, we verify that the equation is correct by computing its associated Picard-Fuchs equation. We also notice that each point of the Teichmüller curve has a hyperflex and we see that the torsion map is a central projection from this point.

Keywords:  Teichmüller curve, Veech group is triangle group, universal family of curves, Picard-Fuchs equation.
Mathematics Subject Classification:  Primary: 14H10, 32G15, 14H30.

Received: February 2016;      Revised: August 2016;      Available Online: December 2016.

 References