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. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

\bysame, Limit theorems for translation flows,, (2010), (2010), 1. Google Scholar

[9]

D. Chen, Covers of the projective line and the moduli space of quadratic differentials,, (2010), (2010), 1. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[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. doi: 10.2307/2952464. Google Scholar

[14]

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

[15]

\bysame, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, in, 1B (2006), 549. Google Scholar

[16]

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

[17]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum,, 2008, (): 1. Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1996), 318. Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

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

[28]

C. McMullen, Braid groups and Hodge theory,, to appear in Math. Ann., (). Google Scholar

[29]

J. Milnor, Remarks concerning spin manifolds,, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), (1965), 55. Google Scholar

[30]

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

[31]

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

[32]

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

[33]

R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials,, 2010, (): 1. Google Scholar

[34]

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

[35]

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

[36]

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

[37]

A. Wright, Abelian square-tiled surfaces,, preprint, (2011). Google Scholar

[38]

A. Zorich, Asymptotic flag of an orientable measured foliation on a surface,, in, (1993), 479. Google Scholar

[39]

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

[40]

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

[41]

\bysame, Square-tiled surfaces and Teichmüller volumes of the moduli spaces of Abelian differentials,, Rigidity in Dynamics and Geometry, (2000), 459. Google Scholar

[42]

\bysame, How do the leaves of a closed $1$-form wind around a surface,, in the collection:, 197 (1999), 135. Google Scholar

[43]

\bysame, Flat surfaces,, in collection, (2003), 9. Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

\bysame, Limit theorems for translation flows,, (2010), (2010), 1. Google Scholar

[9]

D. Chen, Covers of the projective line and the moduli space of quadratic differentials,, (2010), (2010), 1. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[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. doi: 10.2307/2952464. Google Scholar

[14]

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

[15]

\bysame, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, in, 1B (2006), 549. Google Scholar

[16]

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

[17]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum,, 2008, (): 1. Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1996), 318. Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

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

[28]

C. McMullen, Braid groups and Hodge theory,, to appear in Math. Ann., (). Google Scholar

[29]

J. Milnor, Remarks concerning spin manifolds,, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), (1965), 55. Google Scholar

[30]

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

[31]

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

[32]

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

[33]

R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials,, 2010, (): 1. Google Scholar

[34]

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

[35]

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

[36]

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

[37]

A. Wright, Abelian square-tiled surfaces,, preprint, (2011). Google Scholar

[38]

A. Zorich, Asymptotic flag of an orientable measured foliation on a surface,, in, (1993), 479. Google Scholar

[39]

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

[40]

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

[41]

\bysame, Square-tiled surfaces and Teichmüller volumes of the moduli spaces of Abelian differentials,, Rigidity in Dynamics and Geometry, (2000), 459. Google Scholar

[42]

\bysame, How do the leaves of a closed $1$-form wind around a surface,, in the collection:, 197 (1999), 135. Google Scholar

[43]

\bysame, Flat surfaces,, in collection, (2003), 9. Google Scholar

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