Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces

Abstract

We consider normal covers of $\mathbb{C}P^1$ with abelian deck group and branchedover at most four points. Families of such covers yield arithmeticTeichmüller curves, whose period mapping may be describedgeometrically in terms of Schwarz triangle mappings. TheseTeichmüller curves are generated by abelian square-tiled surfaces. We compute all individual Lyapunov exponents for abelian square-tiledsurfaces, and demonstrate a direct and transparent dependence on thegeometry of the period mapping. For this we develop a result ofindependent interest, which, for certain rank two bundles, expressesLyapunov exponents in terms of the period mapping. In the case ofabelian square-tiled surfaces, the Lyapunov exponents are ratios ofareas of hyperbolic triangles.