April  2011, 5(2): 285-318. doi: 10.3934/jmd.2011.5.285

Square-tiled cyclic covers

1. 

Department of Mathematics, University of Maryland, College Park, MD 20742-4015, United States

2. 

Collège de France, 3 Rue d’Ulm, Paris, CEDEX 05, France

3. 

IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes cedex

Received  July 2010 Revised  June 2011 Published  July 2011

A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichmüller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichmüller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in [17] of a Teichmüller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example in genus three found previously by Forni also corresponds to a square-tiled cyclic cover [15]. We present several new examples of Teichmüller curves in strata of holomorphic and meromorphic quadratic differentials with a maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichmüller curves with maximally degenerate spectra. We prove that this is indeed the case within the class of square-tiled cyclic covers.
Citation: Giovanni Forni, Carlos Matheus, Anton Zorich. Square-tiled cyclic covers. Journal of Modern Dynamics, 2011, 5 (2) : 285-318. doi: 10.3934/jmd.2011.5.285
References:
[1]

M. Atiyah, Riemann surfaces and spin structures, Ann. scient. ÉNS (4), 4 (1971), 47-62.

[2]

A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation flows, Ann. of Math. (2), 165 (2007), 637-664. doi: 10.4007/annals.2007.165.637.

[3]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56. doi: 10.1007/s11511-007-0012-1.

[4]

M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geometry and Topology, 11 (2007), 1887-2073. doi: 10.2140/gt.2007.11.1887.

[5]

I. Bouw, The $p$-rank of ramified covers of curves, Compositio Math., 126 (2001), 295-322. doi: 10.1023/A:1017513122376.

[6]

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.

[7]

A. Bufetov, Hölder cocycles and ergodic integrals for translation flows on flat surfaces, Electron. Res. Announc. Math. Sci., 17 (2010), 34-42. doi: 10.3934/era.2010.17.34.

[8]

\bysame, Limit theorems for translation flows, (2010), 1-69, arXiv:0804.3970v3.

[9]

D. Chen, Covers of the projective line and the moduli space of quadratic differentials, (2010), 1-19, arXiv:1005.3120v1.

[10]

A. Eskin and A. Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Inventiones Mathematicae, 145 (2001), 59-103.

[11]

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

[12]

\bysame, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 319-353. doi: 10.3934/jmd.2011.5.319.

[13]

G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math. (2), 146 (1997), 295-344. doi: 10.2307/2952464.

[14]

\bysame, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103. doi: doi:10.2307/3062150.

[15]

\bysame, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in "Handbook of Dynamical Systems" (eds. B. Hasselblatt and A. Katok), 1B, Elsevier B. V., Amsterdam, 2006, 549-580.

[16]

\bysame, A geometric criterion for the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle, J. Mod. Dyn., 5 (2011), 355-395. doi: 10.3934/jmd.2011.5.355.

[17]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum, 2008, 1-8, arXiv:0810.0023v1.

[18]

G. Forni, C. Matheus and A. Zorich, Lyapunov spectrum of equivariant subbundles of the Hodge bundle, in preparation.

[19]

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.

[20]

F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237. doi: 10.1002/mana.200510597.

[21]

D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2), 22 (1980), 365-373. doi: 10.1112/jlms/s2-22.2.365.

[22]

M. Kontsevich, Lyapunov exponents and Hodge theory, in "The Mathematical Beauty of Physics," (Saclay, 1996), Adv. Ser. Math. Phys., 24, World Scientific Publ., River Edge, NJ, 1997, 318-332.

[23]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Inventiones Mathematicae, 153 (2003), 631-678.

[24]

E. Lanneau, Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities, Comment. Math. Helvetici, 79 (2004), 471-501. doi: 10.1007/s00014-004-0806-0.

[25]

P. Lochak, On arithmetic curves in the moduli spaces of curves, J. Inst. Math. Jussieu, 4 (2005), 443-508. doi: 10.1017/S1474748005000101.

[26]

H. Masur and J. Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Comment. Math. Helvetici, 68 (1993), 289-307. doi: 10.1007/BF02565820.

[27]

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.

[28]

C. McMullen, Braid groups and Hodge theory, to appear in Math. Ann.

[29]

J. Milnor, Remarks concerning spin manifolds, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, NJ, 1965, 55-62.

[30]

M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1.

[31]

D. Mumford, Theta-characteristics of an algebraic curve, Ann. scient. Éc. Norm. Sup. (4), 4 (1971), 181-191.

[32]

G. Schmithüsen, An algorithm for finding the Veech group of an origami, Experimental Mathematics, 13 (2004), 459-472.

[33]

R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials, 2010, 1-25, arXiv:1010.1038v2.

[34]

W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242.

[35]

\bysame, The Teichmüller Geodesic Flow, Ann. of Math. (2), 124 (1986), 441-530. doi: 10.2307/2007091.

[36]

\bysame, Moduli spaces of quadratic differentials, J. Anal. Math., 55 (1990), 117-171. doi: 10.1007/BF02789200.

[37]

A. Wright, Abelian square-tiled surfaces, preprint, 2011.

[38]

A. Zorich, Asymptotic flag of an orientable measured foliation on a surface, in "Geometric Study of Foliations," (Tokyo, 1993), World Scientific Publ., River Edge, NJ, 1994, 479-498.

[39]

\bysame, Finite Gauss measure on the space of interval-exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), 46 (1996), 325-370.

[40]

\bysame, Deviation for interval-exchange transformations, Ergod. Th. & Dynam. Sys., 17 (1997), 1477-1499.

[41]

\bysame, Square-tiled surfaces and Teichmüller volumes of the moduli spaces of Abelian differentials, "Rigidity in Dynamics and Geometry" (eds. M. Burger and A. Iozzi), Proceedings of the Programme on Ergodic Theory, Geometric Rigidity and Number Theory, Isaac Newton Institute for the Mathematical Sciences, Cambridge, UK, January 5-July 7, 2000, Springer-Verlag, Berlin, 2002, 459-471.

[42]

\bysame, How do the leaves of a closed $1$-form wind around a surface, in the collection: "Pseudoperiodic Topology" (eds. Vladimir Arnold, Maxim Kontsevich and Anton Zorich), AMS Translations, Ser. 2, 197, Advances in the Mathematical Sciences, 46, AMS, Providence, RI, 1999, 135-178.

[43]

\bysame, Flat surfaces, in collection "Frontiers in Number Theory, Physics and Geometry. Vol. 1: On Random Matrices, Zeta Functions and Dynamical Systems" (eds. P. Cartier, B. Julia, P. Moussa and P. Vanhove), Ecole de physique des Houches, France, March 9-21, 2003, Springer-Verlag, Berlin, 2006, 439-586.

show all references

References:
[1]

M. Atiyah, Riemann surfaces and spin structures, Ann. scient. ÉNS (4), 4 (1971), 47-62.

[2]

A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation flows, Ann. of Math. (2), 165 (2007), 637-664. doi: 10.4007/annals.2007.165.637.

[3]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56. doi: 10.1007/s11511-007-0012-1.

[4]

M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geometry and Topology, 11 (2007), 1887-2073. doi: 10.2140/gt.2007.11.1887.

[5]

I. Bouw, The $p$-rank of ramified covers of curves, Compositio Math., 126 (2001), 295-322. doi: 10.1023/A:1017513122376.

[6]

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.

[7]

A. Bufetov, Hölder cocycles and ergodic integrals for translation flows on flat surfaces, Electron. Res. Announc. Math. Sci., 17 (2010), 34-42. doi: 10.3934/era.2010.17.34.

[8]

\bysame, Limit theorems for translation flows, (2010), 1-69, arXiv:0804.3970v3.

[9]

D. Chen, Covers of the projective line and the moduli space of quadratic differentials, (2010), 1-19, arXiv:1005.3120v1.

[10]

A. Eskin and A. Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Inventiones Mathematicae, 145 (2001), 59-103.

[11]

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

[12]

\bysame, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 319-353. doi: 10.3934/jmd.2011.5.319.

[13]

G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math. (2), 146 (1997), 295-344. doi: 10.2307/2952464.

[14]

\bysame, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103. doi: doi:10.2307/3062150.

[15]

\bysame, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in "Handbook of Dynamical Systems" (eds. B. Hasselblatt and A. Katok), 1B, Elsevier B. V., Amsterdam, 2006, 549-580.

[16]

\bysame, A geometric criterion for the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle, J. Mod. Dyn., 5 (2011), 355-395. doi: 10.3934/jmd.2011.5.355.

[17]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum, 2008, 1-8, arXiv:0810.0023v1.

[18]

G. Forni, C. Matheus and A. Zorich, Lyapunov spectrum of equivariant subbundles of the Hodge bundle, in preparation.

[19]

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.

[20]

F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237. doi: 10.1002/mana.200510597.

[21]

D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2), 22 (1980), 365-373. doi: 10.1112/jlms/s2-22.2.365.

[22]

M. Kontsevich, Lyapunov exponents and Hodge theory, in "The Mathematical Beauty of Physics," (Saclay, 1996), Adv. Ser. Math. Phys., 24, World Scientific Publ., River Edge, NJ, 1997, 318-332.

[23]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Inventiones Mathematicae, 153 (2003), 631-678.

[24]

E. Lanneau, Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities, Comment. Math. Helvetici, 79 (2004), 471-501. doi: 10.1007/s00014-004-0806-0.

[25]

P. Lochak, On arithmetic curves in the moduli spaces of curves, J. Inst. Math. Jussieu, 4 (2005), 443-508. doi: 10.1017/S1474748005000101.

[26]

H. Masur and J. Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Comment. Math. Helvetici, 68 (1993), 289-307. doi: 10.1007/BF02565820.

[27]

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.

[28]

C. McMullen, Braid groups and Hodge theory, to appear in Math. Ann.

[29]

J. Milnor, Remarks concerning spin manifolds, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, NJ, 1965, 55-62.

[30]

M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1.

[31]

D. Mumford, Theta-characteristics of an algebraic curve, Ann. scient. Éc. Norm. Sup. (4), 4 (1971), 181-191.

[32]

G. Schmithüsen, An algorithm for finding the Veech group of an origami, Experimental Mathematics, 13 (2004), 459-472.

[33]

R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials, 2010, 1-25, arXiv:1010.1038v2.

[34]

W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242.

[35]

\bysame, The Teichmüller Geodesic Flow, Ann. of Math. (2), 124 (1986), 441-530. doi: 10.2307/2007091.

[36]

\bysame, Moduli spaces of quadratic differentials, J. Anal. Math., 55 (1990), 117-171. doi: 10.1007/BF02789200.

[37]

A. Wright, Abelian square-tiled surfaces, preprint, 2011.

[38]

A. Zorich, Asymptotic flag of an orientable measured foliation on a surface, in "Geometric Study of Foliations," (Tokyo, 1993), World Scientific Publ., River Edge, NJ, 1994, 479-498.

[39]

\bysame, Finite Gauss measure on the space of interval-exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), 46 (1996), 325-370.

[40]

\bysame, Deviation for interval-exchange transformations, Ergod. Th. & Dynam. Sys., 17 (1997), 1477-1499.

[41]

\bysame, Square-tiled surfaces and Teichmüller volumes of the moduli spaces of Abelian differentials, "Rigidity in Dynamics and Geometry" (eds. M. Burger and A. Iozzi), Proceedings of the Programme on Ergodic Theory, Geometric Rigidity and Number Theory, Isaac Newton Institute for the Mathematical Sciences, Cambridge, UK, January 5-July 7, 2000, Springer-Verlag, Berlin, 2002, 459-471.

[42]

\bysame, How do the leaves of a closed $1$-form wind around a surface, in the collection: "Pseudoperiodic Topology" (eds. Vladimir Arnold, Maxim Kontsevich and Anton Zorich), AMS Translations, Ser. 2, 197, Advances in the Mathematical Sciences, 46, AMS, Providence, RI, 1999, 135-178.

[43]

\bysame, Flat surfaces, in collection "Frontiers in Number Theory, Physics and Geometry. Vol. 1: On Random Matrices, Zeta Functions and Dynamical Systems" (eds. P. Cartier, B. Julia, P. Moussa and P. Vanhove), Ecole de physique des Houches, France, March 9-21, 2003, Springer-Verlag, Berlin, 2006, 439-586.

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