Geodesic
Teichmüller curves, triangle groups, and Lyapunov exponents
Irene I. Bouw1 ; Martin Möller 2
1 Institute of Pure Mathematics, Ulm University, D-89069, Ulm, Germany
2 Institut für Mathematik, Goethe-Universität, Robert-Mayer-Str. 6-8, D-60054 Frankfurt am Main, Germany
Annals of mathematics, Tome 172 (2010) no. 1, pp. 139-185.

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We construct a Teichmüller curve uniformized by a Fuchsian triangle group commensurable to $\Delta(m,n,\infty)$ for every $m,n\leq \infty$. In most cases, for example when $m\neq n$ and $m$ or $n$ is odd, the uniformizing group is equal to the triangle group $\Delta(m,n,\infty)$. Our construction includes the Teichmüller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. For small $m$, we find billiard tables that generate these Teichmüller curves. We interpret some of the so-called Lyapunov exponents of the Kontsevich-Zorich cocycle as normalized degrees of a natural line bundle on a Teichmüller curve. We determine the Lyapunov exponents for the Teichmüller curves we construct.
DOI : 10.4007/annals.2010.172.139

Irene I. Bouw 1 ; Martin Möller  2

1 Institute of Pure Mathematics, Ulm University, D-89069, Ulm, Germany
2 Institut für Mathematik, Goethe-Universität, Robert-Mayer-Str. 6-8, D-60054 Frankfurt am Main, Germany 
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Irene I. Bouw; Martin Möller . Teichmüller curves, triangle groups, and Lyapunov exponents. Annals of mathematics, Tome 172 (2010) no. 1, pp. 139-185. doi : 10.4007/annals.2010.172.139. http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.139/

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