Schwarz三角映射与Teichmüller曲线：Veech-Ward-Bouw-Möller曲线
Schwarz triangle mappings and Teichmüller curves: the Veech-Ward-Bouw-Möller curves
摘要：
本文研究了一类由Bouw和Möller构造的Teichmüller曲线T（n，m），以及Veech和Ward在n= 2，3的情形下构造的曲线族.我们简化了T（n，m）是Teichmüller曲线的证明，避免了用最大Higgs丛来刻画Teichmüller曲线.我们的关键工具是描述的周期映射T（n，m）的Schwarz三角形映射。我们证明了T（n，m）总是由Hooper格面半正多边形分解生成的.我们计算李雅普诺夫指数，并确定在所有情况下的代数连续性。我们证明了T（n，m）上的每个点（黎曼曲面）经常覆盖某个不同T（n '，m'）上的某个点。T（n，m）作为CP ^1的阿贝尔覆盖族的纤维状分支在四个点上分支而出现。CP ^1的这些覆盖可以被认为是阿贝尔平行四边形拼接曲面，这一观点为我们的研究提供了便利。
英文摘要：
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^1 branched over four points. These covers of CP^1 can be considered as abelian parallelogram-tiled surfaces, and this viewpoint facilitates much of our study.
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