American Institute of Mathematical Sciences

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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

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