American Institute of Mathematical Sciences

Advanced
2015, 9: 123-140. doi: 10.3934/jmd.2015.9.123

The relative cohomology of abelian covers of the flat pillowcase

Chenxi Wu 1,

1. 

Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, NY 14853, United States

Received  November 2014 Revised  March 2015 Published  June 2015

We calculate the action of the group of affine diffeomorphisms on the relative cohomology of pair $(M,\Sigma)$, where $M$ is a square-tiled surface that is a normal abelian cover of the flat pillowcase. As an application, we answer a question raised by Smillie and Weiss.
Keywords: Teichmüller dynamics., Square-tiled translation surface.
Mathematics Subject Classification: Primary: 37D40; Secondary: 37E3.
Citation: Chenxi Wu. The relative cohomology of abelian covers of the flat pillowcase. Journal of Modern Dynamics, 2015, 9: 123-140. doi: 10.3934/jmd.2015.9.123
References:
[1]

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.  Google Scholar

[2]

H. S. M. Coxeter, Regular Polytopes, Third edition, Dover Publications, Inc., New York, 1973.  Google Scholar

[3]

L. E. Dickson, Algebraic Theories, Dover Publications, Inc., New York, 1959.  Google Scholar

[4]

P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math., 63 (1986), 5-89.  Google Scholar

[5]

A. Eskin, M. Konstevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, arXiv:1007.5330, 2011. Google Scholar

[6]

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

[7]

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.  Google Scholar

[8]

G. Forni, 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, Elsevier, 2006. Google Scholar

[9]

A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.  Google Scholar

[10]

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

[11]

P. Hubert and G. Schmithüsen, Action of the affine group on cyclic covers,, in preparation., ().   Google Scholar

[12]

C. T. McMullen, Braid groups and Hodge theory, Math. Ann., 355 (2013), 893-946. doi: 10.1007/s00208-012-0804-2.  Google Scholar

[13]

C. Matheus and J.-C. Yoccoz, The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., 4 (2010), 453-486. doi: 10.3934/jmd.2010.4.453.  Google Scholar

[14]

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

[15]

J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[16]

J. Smillie and B. Weiss, Examples of horocycle-invariant measures on the moduli space of translation, surfaces., ().   Google Scholar

[17]

W. P. Thurston, Shapes of polyhedra and triangulations of the sphere, in The Epstein Birthday Schrift, Geometry and Topology Monographs, 1, Geom. Topol. Publ., Coventry, 1998, 511-549. doi: 10.2140/gtm.1998.1.511.  Google Scholar

[18]

A. Wright, Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces, J. Mod. Dyn., 6 (2012), 405-426. doi: 10.3934/jmd.2012.6.405.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

H. S. M. Coxeter, Regular Polytopes, Third edition, Dover Publications, Inc., New York, 1973.  Google Scholar

[3]

L. E. Dickson, Algebraic Theories, Dover Publications, Inc., New York, 1959.  Google Scholar

[4]

P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math., 63 (1986), 5-89.  Google Scholar

[5]

A. Eskin, M. Konstevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, arXiv:1007.5330, 2011. Google Scholar

[6]

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

[7]

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.  Google Scholar

[8]

G. Forni, 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, Elsevier, 2006. Google Scholar

[9]

A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.  Google Scholar

[10]

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

[11]

P. Hubert and G. Schmithüsen, Action of the affine group on cyclic covers,, in preparation., ().   Google Scholar

[12]

C. T. McMullen, Braid groups and Hodge theory, Math. Ann., 355 (2013), 893-946. doi: 10.1007/s00208-012-0804-2.  Google Scholar

[13]

C. Matheus and J.-C. Yoccoz, The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., 4 (2010), 453-486. doi: 10.3934/jmd.2010.4.453.  Google Scholar

[14]

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

[15]

J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[16]

J. Smillie and B. Weiss, Examples of horocycle-invariant measures on the moduli space of translation, surfaces., ().   Google Scholar

[17]

W. P. Thurston, Shapes of polyhedra and triangulations of the sphere, in The Epstein Birthday Schrift, Geometry and Topology Monographs, 1, Geom. Topol. Publ., Coventry, 1998, 511-549. doi: 10.2140/gtm.1998.1.511.  Google Scholar

[18]

A. Wright, Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces, J. Mod. Dyn., 6 (2012), 405-426. doi: 10.3934/jmd.2012.6.405.  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, Carlos Matheus, Anton Zorich. Square-tiled cyclic covers. Journal of Modern Dynamics, 2011, 5 (2) : 285-318. doi: 10.3934/jmd.2011.5.285
[3]

Sébastien Ferenczi, Pascal Hubert. Rigidity of square-tiled interval exchange transformations. Journal of Modern Dynamics, 2019, 14: 153-177. doi: 10.3934/jmd.2019006
[4]

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
[5]

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
[6]

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
[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]

Martin Möller. Shimura and Teichmüller curves. Journal of Modern Dynamics, 2011, 5 (1) : 1-32. doi: 10.3934/jmd.2011.5.1
[9]

Giovanni Forni. On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations. Journal of Modern Dynamics, 2012, 6 (2) : 139-182. doi: 10.3934/jmd.2012.6.139
[10]

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
[11]

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
[12]

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
[13]

David Aulicino, Chaya Norton. Shimura–Teichmüller curves in genus 5. Journal of Modern Dynamics, 2020, 16: 255-288. doi: 10.3934/jmd.2020009
[14]

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
[15]

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
[16]

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
[17]

Anna Lenzhen, Babak Modami, Kasra Rafi. Teichmüller geodesics with $ d$-dimensional limit sets. Journal of Modern Dynamics, 2018, 12: 261-283. doi: 10.3934/jmd.2018010
[18]

Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic & Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025
[19]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485
[20]

W. Patrick Hooper. An infinite surface with the lattice property Ⅱ: Dynamics of pseudo-Anosovs. Journal of Modern Dynamics, 2019, 14: 243-276. doi: 10.3934/jmd.2019009

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences