Teichmüller curves, triangle groups, and Lyapunov exponents

Abstract

We construct a Teichmüller curve uniformized by a Fuchsian triangle group commensurable to Δ(m;n; ∞) for every m,n < ∞. In most cases, for example when m≠n and m or n is odd, the uniformizing group is equal to the triangle group Δ(m;n;∞). 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. © 2010 Annals of Mathematics.