Schwarz triangle mappings and Teichmüller curves: the Veech–Ward–Bouw–Möller curves

Abstract

We study a family of Teichmuller curves \({\mathcal{T}\,(n,m)}\) constructed by Bouw and Moller, and previously by Veech and Ward in the cases n = 2,3. We simplify the proof that \({\mathcal{T}\,(n,m)}\) is a Teichmuller curve, avoiding the use Moller’s characterization of Teichmuller curves in terms of maximally Higgs bundles. Our key tool is a description of the period mapping of \({\mathcal{T}\,(n,m)}\) in terms of Schwarz triangle mappings. We prove that \({\mathcal{T}\,(n,m)}\) 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 \({\mathcal{T}\,(n,m)}\) covers some point on some distinct \({\mathcal{T}\,(n',m').}\) The \({\mathcal{T}\,(n,m)}\) arise as fiberwise quotients of families of abelian covers of \({\mathbb{C}{\rm P^{1}}}\) branched over four points. These covers of \({\mathbb{C}{\rm P^{1}}}\) can be considered as abelian parallelogram-tiled surfaces, and this viewpoint facilitates much of our study.