# American Institute of Mathematical Sciences

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] I. Bouw, The $p$-rank of ramified covers of curves, Compositio Math., 126 (2001), 295-322. 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), 139-185. 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), 34-42. doi: 10.3934/era.2010.17.34.  Google Scholar [8] \bysame, Limit theorems for translation flows, (2010), 1-69, arXiv:0804.3970v3. Google Scholar [9] D. Chen, Covers of the projective line and the moduli space of quadratic differentials, (2010), 1-19, arXiv:1005.3120v1. 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), 59-103.  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), 319-353. 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), 295-344. 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), 1-103. doi: doi:10.2307/3062150.  Google Scholar [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.  Google Scholar [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.  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), 191-213. doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar [20] F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237. doi: 10.1002/mana.200510597.  Google Scholar [21] D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2), 22 (1980), 365-373. doi: 10.1112/jlms/s2-22.2.365.  Google Scholar [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.  Google Scholar [23] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Inventiones Mathematicae, 153 (2003), 631-678.  Google Scholar [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.  Google Scholar [25] P. Lochak, On arithmetic curves in the moduli spaces of curves, J. Inst. Math. Jussieu, 4 (2005), 443-508. 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), 289-307. 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), 453-486. 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), Princeton Univ. Press, Princeton, NJ, 1965, 55-62.  Google Scholar [30] M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. 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), 181-191.  Google Scholar [32] G. Schmithüsen, An algorithm for finding the Veech group of an origami, Experimental Mathematics, 13 (2004), 459-472.  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), 201-242.  Google Scholar [35] \bysame, The Teichmüller Geodesic Flow, Ann. of Math. (2), 124 (1986), 441-530. doi: 10.2307/2007091.  Google Scholar [36] \bysame, Moduli spaces of quadratic differentials, J. Anal. Math., 55 (1990), 117-171. 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 "Geometric Study of Foliations," (Tokyo, 1993), World Scientific Publ., River Edge, NJ, 1994, 479-498.  Google Scholar [39] \bysame, Finite Gauss measure on the space of interval-exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), 46 (1996), 325-370.  Google Scholar [40] \bysame, Deviation for interval-exchange transformations, Ergod. Th. & Dynam. Sys., 17 (1997), 1477-1499.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### References:
 [1] M. Atiyah, Riemann surfaces and spin structures, Ann. scient. ÉNS (4), 4 (1971), 47-62.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] I. Bouw, The $p$-rank of ramified covers of curves, Compositio Math., 126 (2001), 295-322. 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), 139-185. 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), 34-42. doi: 10.3934/era.2010.17.34.  Google Scholar [8] \bysame, Limit theorems for translation flows, (2010), 1-69, arXiv:0804.3970v3. Google Scholar [9] D. Chen, Covers of the projective line and the moduli space of quadratic differentials, (2010), 1-19, arXiv:1005.3120v1. 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), 59-103.  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), 319-353. 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), 295-344. 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), 1-103. doi: doi:10.2307/3062150.  Google Scholar [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.  Google Scholar [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.  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), 191-213. doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar [20] F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237. doi: 10.1002/mana.200510597.  Google Scholar [21] D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2), 22 (1980), 365-373. doi: 10.1112/jlms/s2-22.2.365.  Google Scholar [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.  Google Scholar [23] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Inventiones Mathematicae, 153 (2003), 631-678.  Google Scholar [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.  Google Scholar [25] P. Lochak, On arithmetic curves in the moduli spaces of curves, J. Inst. Math. Jussieu, 4 (2005), 443-508. 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), 289-307. 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), 453-486. 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), Princeton Univ. Press, Princeton, NJ, 1965, 55-62.  Google Scholar [30] M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. 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), 181-191.  Google Scholar [32] G. Schmithüsen, An algorithm for finding the Veech group of an origami, Experimental Mathematics, 13 (2004), 459-472.  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), 201-242.  Google Scholar [35] \bysame, The Teichmüller Geodesic Flow, Ann. of Math. (2), 124 (1986), 441-530. doi: 10.2307/2007091.  Google Scholar [36] \bysame, Moduli spaces of quadratic differentials, J. Anal. Math., 55 (1990), 117-171. 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 "Geometric Study of Foliations," (Tokyo, 1993), World Scientific Publ., River Edge, NJ, 1994, 479-498.  Google Scholar [39] \bysame, Finite Gauss measure on the space of interval-exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), 46 (1996), 325-370.  Google Scholar [40] \bysame, Deviation for interval-exchange transformations, Ergod. Th. & Dynam. Sys., 17 (1997), 1477-1499.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] 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 [2] Giovanni Forni. A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle. Journal of Modern Dynamics, 2011, 5 (2) : 355-395. doi: 10.3934/jmd.2011.5.355 [3] Artur Avila, Carlos Matheus, Jean-Christophe Yoccoz. The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces. Journal of Modern Dynamics, 2019, 14: 21-54. doi: 10.3934/jmd.2019002 [4] Rodolfo Gutiérrez-Romo. A family of quaternionic monodromy groups of the Kontsevich–Zorich cocycle. Journal of Modern Dynamics, 2019, 14: 227-242. doi: 10.3934/jmd.2019008 [5] Sébastien Ferenczi, Pascal Hubert. Rigidity of square-tiled interval exchange transformations. Journal of Modern Dynamics, 2019, 14: 153-177. doi: 10.3934/jmd.2019006 [6] Alex Eskin, Maxim Kontsevich, Anton Zorich. Lyapunov spectrum of square-tiled cyclic covers. Journal of Modern Dynamics, 2011, 5 (2) : 319-353. doi: 10.3934/jmd.2011.5.319 [7] Francisco Arana-Herrera. Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics. Journal of Modern Dynamics, 2020, 16: 81-107. doi: 10.3934/jmd.2020004 [8] Ursula Hamenstädt. Bowen's construction for the Teichmüller flow. Journal of Modern Dynamics, 2013, 7 (4) : 489-526. doi: 10.3934/jmd.2013.7.489 [9] Ursula Hamenstädt. Dynamics of the Teichmüller flow on compact invariant sets. Journal of Modern Dynamics, 2010, 4 (2) : 393-418. doi: 10.3934/jmd.2010.4.393 [10] Martin Möller. Shimura and Teichmüller curves. Journal of Modern Dynamics, 2011, 5 (1) : 1-32. doi: 10.3934/jmd.2011.5.1 [11] Giovanni Forni, Carlos Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. Journal of Modern Dynamics, 2014, 8 (3&4) : 271-436. doi: 10.3934/jmd.2014.8.271 [12] Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581 [13] Jon Fickenscher. A combinatorial proof of the Kontsevich-Zorich-Boissy classification of Rauzy classes. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1983-2025. doi: 10.3934/dcds.2016.36.1983 [14] Dawei Chen. Strata of abelian differentials and the Teichmüller dynamics. Journal of Modern Dynamics, 2013, 7 (1) : 135-152. doi: 10.3934/jmd.2013.7.135 [15] Fei Yu, Kang Zuo. Weierstrass filtration on Teichmüller curves and Lyapunov exponents. Journal of Modern Dynamics, 2013, 7 (2) : 209-237. doi: 10.3934/jmd.2013.7.209 [16] David Aulicino, Chaya Norton. Shimura–Teichmüller curves in genus 5. Journal of Modern Dynamics, 2020, 16: 255-288. doi: 10.3934/jmd.2020009 [17] Guizhen Cui, Yunping Jiang, Anthony Quas. Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 535-552. doi: 10.3934/dcds.1999.5.535 [18] Chi Po Choi, Xianfeng Gu, Lok Ming Lui. Subdivision connectivity remeshing via Teichmüller extremal map. Inverse Problems & Imaging, 2017, 11 (5) : 825-855. doi: 10.3934/ipi.2017039 [19] Matteo Costantini, André Kappes. The equation of the Kenyon-Smillie (2, 3, 4)-Teichmüller curve. Journal of Modern Dynamics, 2017, 11: 17-41. doi: 10.3934/jmd.2017002 [20] Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147

2020 Impact Factor: 0.848