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Schwarz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller curves. (English) Zbl 1267.30099
Summary: We study a family of Teichmüller curves \(\mathcal T(n,m)\) constructed by I. Bouw and M. Möller [Ann. Math. (2) 172, No. 1, 139–185 (2010; Zbl 1203.37049)], and previously by W. A. Veech [Invent. Math. 97, No. 3, 553–583 (1989; Zbl 0676.32006); erratum ibid. 103, No. 2, 447 (1991; Zbl 0709.32014)] and C. C. Ward [Ergodic Theory Dyn. Syst. 18, No. 4, 1019–1042 (1998; Zbl 0915.58059)] 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 \(\mathcal T(n^\prime,m^\prime)\). The \(\mathcal T(n,m)\) arise as fiberwise quotients of families of abelian covers of \(\mathbb C\mathrm P^1\) branched over four points. These covers of \(\mathbb C\mathrm P^1\) can be considered as abelian parallelogram-tiled surfaces, and this viewpoint facilitates much of our study.

MSC:
30F10 Compact Riemann surfaces and uniformization
30F30 Differentials on Riemann surfaces
30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
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