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

摘要

We study a family of Teichmüller curves ({mathcal{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 ({mathcal{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 ({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.