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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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##### References:
 [1] Bouw, I.I.; Möller, M., Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2), 172, 139-185, (2010) · Zbl 1203.37049 [2] Bainbridge, M.; Möller, M., The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3, Acta Mathematica, 208, 1-92, (2012) · Zbl 1250.14014 [3] Calta, K., Veech surfaces and complete periodicity in genus two, Journal of American Mathematical Society, 17, 871-908, (2004) · Zbl 1073.37032 [4] Carocca, A.; Lange, H.; Rodríguez, R.E., Jacobians with complex multiplication, Transactions of the American Mathematical Society, 363, 6159-6175, (2011) · Zbl 1285.11089 [5] A. Eskin, M. Kontsevich, and A. Zorich, Sum of lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow (preprint). arXiv 1112.5872 (2012). · Zbl 1094.14018 [6] Ellenberg, J.S., Endomorphism algebras of Jacobians, Advances in Mathematics, 162, 243-271, (2001) · Zbl 1065.14507 [7] Forni, G., Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Annals of Mathematics (2), 155, 1-103, (2002) · Zbl 1034.37003 [8] G. Forni. On the Lyapunov exponents of the Kontsevich-Zorich cocycle. In: Handbook of dynamical systems. Vol. 1B, Elsevier, Amsterdam (2006), pp. 549-580. · Zbl 1130.37302 [9] W.P. Hooper. Grid graphs and lattice surfaces, preprint. arXiv:0811.0799 (2009). · Zbl 0676.32006 [10] Hubert, P.; Schmidt, T.A., Invariants of translation surfaces, Ann. Inst. Fourier (Grenoble), 51, 461-495, (2001) · Zbl 0985.32008 [11] J.H. Hubbard. Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, 2006, Teichmüller theory, With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, With forewords by William Thurston and Clifford Earle. · Zbl 1034.37003 [12] Imayoshi Y., Taniguchi M.: An introduction to Teichmüller spaces. Springer, Tokyo (1992) · Zbl 0754.30001 [13] Kenyon, R.; Smillie, J., Billiards on rational-angled triangles, Commentarii Mathematici Helvetici, 75, 65-108, (2000) · Zbl 0967.37019 [14] Lochak, P., On arithmetic curves in the moduli spaces of curves, Journal of Institute of Mathathematics of Jussieu, 4, 443-508, (2005) · Zbl 1094.14018 [15] C.T. McMullen. Billiards and Teichmüller curves on Hilbert modular surfaces. Journal of the American Mathematical Society. (4)16 (2003), 857-885 (electronic). · Zbl 1030.32012 [16] McMullen, C.T., Prym varieties and Teichmüller curves, Duke Mathematical Journal, 133, 569-590, (2006) · Zbl 1099.14018 [17] McMullen, C.T., Teichmüller curves in genus two: torsion divisors and ratios of sines, Inventiones Mathematicae, 165, 651-672, (2006) · Zbl 1103.14014 [18] Möller, M., Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Inventiones Mathematicae, 165, 633-649, (2006) · Zbl 1111.14019 [19] M. Möller. Variations of Hodge structures of a Teichmüller curve. Journal of the American Mathematical Society, (2)19 (2006), 327-344 (electronic). · Zbl 0985.32008 [20] Möller, M., Finiteness results for Teichmüller curves, Annales de l’Institut Fourier (Grenoble), 58, 63-83, (2008) · Zbl 1140.14010 [21] C. Maclachlan and A.W. Reid. The arithmetic of hyperbolic 3-manifolds. Graduate Texts in Mathematics, Vol. 219. Springer, New York (2003). · Zbl 1025.57001 [22] Veech, W.A., Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Inventiones Mathematicae, 97, 553-583, (1989) · Zbl 0676.32006 [23] Ward, C.C., Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergodic Theory and Dynamical Systems, 18, 1019-1042, (1998) · Zbl 0915.58059 [24] Wright, A., Schwarz triangle mappings and Teichmüller curves: abelian square-tiled surfaces, Journal of Modern Dynamics, 6, 405-426, (2012) · Zbl 1254.32021
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