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
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______, Lyapunov spectrum of square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 319. doi: 10.3934/jmd.2011.5.319. Google Scholar

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G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150. Google Scholar

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______, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, in, (2006), 549. Google Scholar

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G. Forni and C. Matheus, An example of a Teichmuller disk in genus 4 with degenerate Kontsevich-Zorich spectrum,, preprint, (2008). Google Scholar

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G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 285. doi: 10.3934/jmd.2011.5.285. Google Scholar

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E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191. doi: 10.1215/S0012-7094-00-10321-3. Google Scholar

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M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1997), 318. Google Scholar

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H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7. Google Scholar

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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. doi: 10.3934/jmd.2010.4.453. Google Scholar

[16]

M. Möller, Teichmüller curves, Galois actions and $\hat{GT}$-relations,, Math. Nachr., 278 (2005), 1061. doi: 10.1002/mana.200310292. Google Scholar

[17]

______, Shimura and Teichmüller curves,, J. Mod. Dyn., 5 (2011), 1. doi: 10.3934/jmd.2011.5.1. Google Scholar

[18]

A. Wright, Schwarz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller curves,, preprint, (2012). Google Scholar

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M. Yoshida, "Fuchsian Differential Equations. With Special Emphasis on the Gauss-Schwarz Theory,", Aspects of Mathematics, E11 (1987). Google Scholar

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A. Zorich, Flat surfaces,, in, (2006), 437. doi: 10.1007/978-3-540-31347-2_13. Google Scholar

show all references

References:
[1]

F. Beukers, Gauss' hypergeometric functions,, , (). Google Scholar

[2]

______, Notes on differential equations and hypergeometric functions,, , (). Google Scholar

[3]

I. I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents,, Ann. of Math. (2), 172 (2010), 139. doi: 10.4007/annals.2010.172.139. Google Scholar

[4]

J. Carlson, S. Müller-Stach and C. Peters, "Period Mappings and Period Domains,", Cambridge Studies in Advanced Mathematics, 85 (2003). Google Scholar

[5]

T. A. Driscoll and L. N. Trefethen, "Schwarz-Christoffel Mapping,", Cambridge Monographs on Applied and Computational Mathematics, 8 (2002). Google Scholar

[6]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, preprint., (). Google Scholar

[7]

______, Lyapunov spectrum of square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 319. doi: 10.3934/jmd.2011.5.319. Google Scholar

[8]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150. Google Scholar

[9]

______, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, in, (2006), 549. Google Scholar

[10]

G. Forni and C. Matheus, An example of a Teichmuller disk in genus 4 with degenerate Kontsevich-Zorich spectrum,, preprint, (2008). Google Scholar

[11]

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 285. doi: 10.3934/jmd.2011.5.285. Google Scholar

[12]

E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191. doi: 10.1215/S0012-7094-00-10321-3. Google Scholar

[13]

M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1997), 318. Google Scholar

[14]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7. Google Scholar

[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. doi: 10.3934/jmd.2010.4.453. Google Scholar

[16]

M. Möller, Teichmüller curves, Galois actions and $\hat{GT}$-relations,, Math. Nachr., 278 (2005), 1061. doi: 10.1002/mana.200310292. Google Scholar

[17]

______, Shimura and Teichmüller curves,, J. Mod. Dyn., 5 (2011), 1. doi: 10.3934/jmd.2011.5.1. Google Scholar

[18]

A. Wright, Schwarz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller curves,, preprint, (2012). Google Scholar

[19]

M. Yoshida, "Fuchsian Differential Equations. With Special Emphasis on the Gauss-Schwarz Theory,", Aspects of Mathematics, E11 (1987). Google Scholar

[20]

A. Zorich, Flat surfaces,, in, (2006), 437. doi: 10.1007/978-3-540-31347-2_13. Google Scholar

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