The relative cohomology of abelian covers of the flat pillowcase

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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

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

Received November 2014 Revised March 2015 Published June 2015

Abstract
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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.
[2]	H. S. M. Coxeter, Regular Polytopes, Third edition, Dover Publications, Inc., New York, 1973.
[3]	L. E. Dickson, Algebraic Theories, Dover Publications, Inc., New York, 1959.
[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.
[5]	A. Eskin, M. Konstevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, arXiv:1007.5330, 2011.
[6]	G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum, arXiv:0810.0023, 2008.
[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.
[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.
[9]	A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
[10]	F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237. doi: 10.1002/mana.200510597.
[11]	P. Hubert and G. Schmithüsen, Action of the affine group on cyclic covers, in preparation.
[12]	C. T. McMullen, Braid groups and Hodge theory, Math. Ann., 355 (2013), 893-946. doi: 10.1007/s00208-012-0804-2.
[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.
[14]	G. Schmithüsen, An algorithm for finding the Veech group of an origami, Experimental Mathematics, 13 (2004), 459-472.
[15]	J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977.
[16]	J. Smillie and B. Weiss, Examples of horocycle-invariant measures on the moduli space of translation surfaces.
[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.
[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.

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.
[2]	H. S. M. Coxeter, Regular Polytopes, Third edition, Dover Publications, Inc., New York, 1973.
[3]	L. E. Dickson, Algebraic Theories, Dover Publications, Inc., New York, 1959.
[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.
[5]	A. Eskin, M. Konstevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, arXiv:1007.5330, 2011.
[6]	G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum, arXiv:0810.0023, 2008.
[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.
[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.
[9]	A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
[10]	F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237. doi: 10.1002/mana.200510597.
[11]	P. Hubert and G. Schmithüsen, Action of the affine group on cyclic covers, in preparation.
[12]	C. T. McMullen, Braid groups and Hodge theory, Math. Ann., 355 (2013), 893-946. doi: 10.1007/s00208-012-0804-2.
[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.
[14]	G. Schmithüsen, An algorithm for finding the Veech group of an origami, Experimental Mathematics, 13 (2004), 459-472.
[15]	J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977.
[16]	J. Smillie and B. Weiss, Examples of horocycle-invariant measures on the moduli space of translation surfaces.
[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.
[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.

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