Schwarz Triangle Mappings and Teichmüller Curves: the Veech-ward-bouw-möller Curves

We study a family of Teichmüller curves T (n,m) constructed by Bouw and Möller, and previously by Veech and Ward in the cases n = 2, 3. We simplify the proof that T (n,m) is a Teichmüller curve, avoiding the use Möller’s characterization of Teichmüller curves in terms of maximally Higgs bundles. Our key tool is a description of the period mapping of T (n,m) in terms of Schwarz triangle mappings. We prove that T (n,m) is always generated by Hooper’s lattice surface with semiregular polygon decomposition. We compute Lyapunov exponents, and determine algebraic primitivity in all cases. We show that frequently, every point (Riemann surface) on T (n,m) covers some point on some distinct T (n′,m′). The T (n,m) arise as fiberwise quotients of families of abelian covers of CP branched over four points. These covers of CP can be considered as abelian parallelogram-tiled surfaces, and this viewpoint facilitates much of our study.