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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.
@article{10_4007_annals_2010_172_139,
author = {Irene I. Bouw and Martin M\"oller },
title = {Teichm\"uller curves, triangle groups, and {Lyapunov} exponents},
journal = {Annals of mathematics},
pages = {139--185},
publisher = {mathdoc},
volume = {172},
number = {1},
year = {2010},
doi = {10.4007/annals.2010.172.139},
mrnumber = {2680418},
zbl = {1203.37049},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.139/}
}
TY - JOUR AU - Irene I. Bouw AU - Martin Möller TI - Teichmüller curves, triangle groups, and Lyapunov exponents JO - Annals of mathematics PY - 2010 SP - 139 EP - 185 VL - 172 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.139/ DO - 10.4007/annals.2010.172.139 LA - en ID - 10_4007_annals_2010_172_139 ER -
%0 Journal Article %A Irene I. Bouw %A Martin Möller %T Teichmüller curves, triangle groups, and Lyapunov exponents %J Annals of mathematics %D 2010 %P 139-185 %V 172 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.139/ %R 10.4007/annals.2010.172.139 %G en %F 10_4007_annals_2010_172_139
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
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