Teichmüller curves, triangle groups, and Lyapunov exponents

Abstract

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.