# American Institute of Mathematical Sciences

July  2012, 6(3): 405-426. doi: 10.3934/jmd.2012.6.405

## Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces

 1 Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637, United States

Received  May 2012 Published  October 2012

We consider normal covers of $\mathbb{C}P^1$ with abelian deck group and branched over at most four points. Families of such covers yield arithmetic Teichmüller curves, whose period mapping may be described geometrically in terms of Schwarz triangle mappings. These Teichmüller curves are generated by abelian square-tiled surfaces.
We compute all individual Lyapunov exponents for abelian square-tiled surfaces, and demonstrate a direct and transparent dependence on the geometry of the period mapping. For this we develop a result of independent interest, which, for certain rank two bundles, expresses Lyapunov exponents in terms of the period mapping. In the case of abelian square-tiled surfaces, the Lyapunov exponents are ratios of areas of hyperbolic triangles.
Citation: Alex Wright. Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces. Journal of Modern Dynamics, 2012, 6 (3) : 405-426. doi: 10.3934/jmd.2012.6.405
##### References:
 [1] F. Beukers, Gauss' hypergeometric functions, http://www.staff.science.uu.nl/~beuke106/MRIcourse93.ps. [2] ______, Notes on differential equations and hypergeometric functions, http://pages.uoregon.edu/njp/beukers.pdf. [3] I. I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2), 172 (2010), 139-185. doi: 10.4007/annals.2010.172.139. [4] J. Carlson, S. Müller-Stach and C. Peters, "Period Mappings and Period Domains," Cambridge Studies in Advanced Mathematics, 85, Cambridge University Press, Cambridge, 2003. [5] T. A. Driscoll and L. N. Trefethen, "Schwarz-Christoffel Mapping," Cambridge Monographs on Applied and Computational Mathematics, 8, Cambridge University Press, Cambridge, 2002. [6] A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, preprint. [7] ______, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 319-353. doi: 10.3934/jmd.2011.5.319. [8] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103. doi: 10.2307/3062150. [9] ______, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in "Handbook of Dynamical Systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 549-580. [10] G. Forni and C. Matheus, An example of a Teichmuller disk in genus 4 with degenerate Kontsevich-Zorich spectrum, preprint, arXiv:0810.0023, 2008. [11] G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318. doi: 10.3934/jmd.2011.5.285. [12] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J., 103 (2000), 191-213. doi: 10.1215/S0012-7094-00-10321-3. [13] M. Kontsevich, Lyapunov exponents and Hodge theory, in "The Mathematical Beauty of Physics" (Saclay, 1996), Adv. Ser. Math. Phys., 24, World Sci. Publ., River Edge, NJ, (1997), 318-332. [14] H. Masur and S. Tabachnikov, Rational billiards and flat structures, in "Handbook of Dynamical Systems," Vol. 1A, North-Holland, Amsterdam, (2002), 1015-1089. doi: 10.1016/S1874-575X(02)80015-7. [15] C. Matheus and J.-C. Yoccoz, The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, Journal of Modern Dynamics, 4 (2010), 453-486. doi: 10.3934/jmd.2010.4.453. [16] M. Möller, Teichmüller curves, Galois actions and $\hat{GT}$-relations, Math. Nachr., 278 (2005), 1061-1077. doi: 10.1002/mana.200310292. [17] ______, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1. [18] A. Wright, Schwarz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller curves, preprint, arXiv:1203.2685, 2012. [19] M. Yoshida, "Fuchsian Differential Equations. With Special Emphasis on the Gauss-Schwarz Theory," Aspects of Mathematics, E11, Friedr. Vieweg & Sohn, Braunschweig, 1987. [20] A. Zorich, Flat surfaces, in "Frontiers in Number Theory, Physics, and Geometry. I," Springer, Berlin, (2006), 437-583. doi: 10.1007/978-3-540-31347-2_13.

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##### References:
 [1] F. Beukers, Gauss' hypergeometric functions, http://www.staff.science.uu.nl/~beuke106/MRIcourse93.ps. [2] ______, Notes on differential equations and hypergeometric functions, http://pages.uoregon.edu/njp/beukers.pdf. [3] I. I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2), 172 (2010), 139-185. doi: 10.4007/annals.2010.172.139. [4] J. Carlson, S. Müller-Stach and C. Peters, "Period Mappings and Period Domains," Cambridge Studies in Advanced Mathematics, 85, Cambridge University Press, Cambridge, 2003. [5] T. A. Driscoll and L. N. Trefethen, "Schwarz-Christoffel Mapping," Cambridge Monographs on Applied and Computational Mathematics, 8, Cambridge University Press, Cambridge, 2002. [6] A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, preprint. [7] ______, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 319-353. doi: 10.3934/jmd.2011.5.319. [8] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103. doi: 10.2307/3062150. [9] ______, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in "Handbook of Dynamical Systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 549-580. [10] G. Forni and C. Matheus, An example of a Teichmuller disk in genus 4 with degenerate Kontsevich-Zorich spectrum, preprint, arXiv:0810.0023, 2008. [11] G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318. doi: 10.3934/jmd.2011.5.285. [12] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J., 103 (2000), 191-213. doi: 10.1215/S0012-7094-00-10321-3. [13] M. Kontsevich, Lyapunov exponents and Hodge theory, in "The Mathematical Beauty of Physics" (Saclay, 1996), Adv. Ser. Math. Phys., 24, World Sci. Publ., River Edge, NJ, (1997), 318-332. [14] H. Masur and S. Tabachnikov, Rational billiards and flat structures, in "Handbook of Dynamical Systems," Vol. 1A, North-Holland, Amsterdam, (2002), 1015-1089. doi: 10.1016/S1874-575X(02)80015-7. [15] C. Matheus and J.-C. Yoccoz, The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, Journal of Modern Dynamics, 4 (2010), 453-486. doi: 10.3934/jmd.2010.4.453. [16] M. Möller, Teichmüller curves, Galois actions and $\hat{GT}$-relations, Math. Nachr., 278 (2005), 1061-1077. doi: 10.1002/mana.200310292. [17] ______, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1. [18] A. Wright, Schwarz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller curves, preprint, arXiv:1203.2685, 2012. [19] M. Yoshida, "Fuchsian Differential Equations. With Special Emphasis on the Gauss-Schwarz Theory," Aspects of Mathematics, E11, Friedr. Vieweg & Sohn, Braunschweig, 1987. [20] A. Zorich, Flat surfaces, in "Frontiers in Number Theory, Physics, and Geometry. I," Springer, Berlin, (2006), 437-583. doi: 10.1007/978-3-540-31347-2_13.
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