Infinite translation surfaces with infinitely generated Veech groups

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October 2010, 4(4): 715-732. doi: 10.3934/jmd.2010.4.715

Infinite translation surfaces with infinitely generated Veech groups

Pascal Hubert ^1, and Gabriela Schmithüsen ^2,

1.	LATP, case cour A, Faculté des sciences Saint Jérôme, Avenue Escadrille Normandie Niemen, 13397 Marseille cedex 20, France
2.	Institute for Algebra and Geometry, University of Karlsruhe, 76128 Karlsruhe, Germany

Received June 2010 Revised September 2010 Published January 2011

Abstract
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We study infinite translation surfaces which are $\ZZ$-covers of finite square-tiled surfaces obtained by a certain two-slit cut and paste construction. We show that if the finite translation surface has a one-cylinder decomposition in some direction, then the Veech group of the infinite translation surface is either a lattice or an infinitely generated group of the first kind. The square-tiled surfaces of genus two with one zero provide examples for finite translation surfaces that fulfill the prerequisites of the theorem.

Keywords: Veech groups, infinite translation surfaces, holomorphic differentials..

Mathematics Subject Classification: 30F35, 30F30, 14H30, 37D5.

Citation: Pascal Hubert, Gabriela Schmithüsen. Infinite translation surfaces with infinitely generated Veech groups. Journal of Modern Dynamics, 2010, 4 (4) : 715-732. doi: 10.3934/jmd.2010.4.715

References:

[1]	J. Bowman, "Flat Structures and Complex Structures in Teichmüller Theory," Thesis (Ph.D.)–Cornell University. ProQuest LLC, Ann Arbor, MI, 2009.
[2]	R. G. Burns and A. M. Brunner, Two remarks on the group property of Howson, Algebra Logic, 18 (1980), 319-325. doi: 10.1007/BF01673500.
[3]	R. Chamanara, Affine automorphism groups of surfaces of infinite type, In the tradition of Ahlfors and Bers, III, 123-145, Contemp. Math., 355, Amer. Math. Soc., Providence, RI, 2004.
[4]	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.
[5]	F. Herrlich, Teichmüller curves defined by characteristic origamis, The geometry of Riemann surfaces and abelian varieties, 133-144, Contemp. Math., 397, Amer. Math. Soc., Providence, RI, 2006.
[6]	W. P. Hooper, Dynamics on an infinite surface with the lattice property, (2007) preprint, arXiv:0802.0189.
[7]	W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry, to appear in Annales de L'Institut Fourier (2009).
[8]	P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $H(2)$, Israel J. Math., 151 (2006), 281-321. doi: 10.1007/BF02777365.
[9]	P. Hubert and S. Lelièvre, Noncongruence subgroups in $H(2)$, Int. Math. Res. Not., (2005), 47-64.
[10]	P. Hubert and T. Schmidt, Infinitely generated Veech groups, Duke Math. J., 123 (2004), 49-69. doi: 10.1215/S0012-7094-04-12312-8.
[11]	P. Hubert and B. Weiss, Dynamics on the infinite staircase, (2008) preprint.
[12]	M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. doi: 10.1007/s00222-003-0303-x.
[13]	S. Lelièvre and R. Silhol, Multi-geodesic tessellations, fractional Dehn twists and uniformization of algebraic curves, (2007) preprint, arXiv:math/0702374.
[14]	C. McMullen, Teichmüller geodesics of infinite complexity, Acta Math., 191 (2003), 191-223. doi: 10.1007/BF02392964.
[15]	P. Przytycki, G. Schmithüsen and F. Valdez, Veech groups of Loch Ness monsters,, to appear in Annales de l'Institut Fourier., ().
[16]	G. Schmithüsen, "Veech Groups of Origamis," Dissertation 2005, Universität Karlsruhe.
[17]	G. Schmithüsen, An algorithm for finding the Veech group of an origami, Experiment. Math., 13 (2004), 459-472.
[18]	G. Schmithüsen, Examples for Veech groups of origamis, The geometry of Riemann surfaces and abelian varieties, 193-206, Contemp. Math., 397, Amer. Math. Soc., Providence, RI, 2006.
[19]	G. Schmithüsen, Origamis with non-congruence Veech groups, Proceedings of 34th Symposium on Transformation Groups, 31-55, Wing Co., Wakayama, 2007.
[20]	F. Valdez, Billiards in polygons and homogeneous foliations on $\CC^2$, Ergod. Th. & Dynam. Sys., 29 (2009), 255-271.
[21]	F. Valdez, Veech groups, irrational billiards and stable abelian differentials, Preprint 2009, arXiv:0905.1591v2.
[22]	W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.

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References:

[1]	J. Bowman, "Flat Structures and Complex Structures in Teichmüller Theory," Thesis (Ph.D.)–Cornell University. ProQuest LLC, Ann Arbor, MI, 2009.
[2]	R. G. Burns and A. M. Brunner, Two remarks on the group property of Howson, Algebra Logic, 18 (1980), 319-325. doi: 10.1007/BF01673500.
[3]	R. Chamanara, Affine automorphism groups of surfaces of infinite type, In the tradition of Ahlfors and Bers, III, 123-145, Contemp. Math., 355, Amer. Math. Soc., Providence, RI, 2004.
[4]	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.
[5]	F. Herrlich, Teichmüller curves defined by characteristic origamis, The geometry of Riemann surfaces and abelian varieties, 133-144, Contemp. Math., 397, Amer. Math. Soc., Providence, RI, 2006.
[6]	W. P. Hooper, Dynamics on an infinite surface with the lattice property, (2007) preprint, arXiv:0802.0189.
[7]	W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry, to appear in Annales de L'Institut Fourier (2009).
[8]	P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $H(2)$, Israel J. Math., 151 (2006), 281-321. doi: 10.1007/BF02777365.
[9]	P. Hubert and S. Lelièvre, Noncongruence subgroups in $H(2)$, Int. Math. Res. Not., (2005), 47-64.
[10]	P. Hubert and T. Schmidt, Infinitely generated Veech groups, Duke Math. J., 123 (2004), 49-69. doi: 10.1215/S0012-7094-04-12312-8.
[11]	P. Hubert and B. Weiss, Dynamics on the infinite staircase, (2008) preprint.
[12]	M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. doi: 10.1007/s00222-003-0303-x.
[13]	S. Lelièvre and R. Silhol, Multi-geodesic tessellations, fractional Dehn twists and uniformization of algebraic curves, (2007) preprint, arXiv:math/0702374.
[14]	C. McMullen, Teichmüller geodesics of infinite complexity, Acta Math., 191 (2003), 191-223. doi: 10.1007/BF02392964.
[15]	P. Przytycki, G. Schmithüsen and F. Valdez, Veech groups of Loch Ness monsters,, to appear in Annales de l'Institut Fourier., ().
[16]	G. Schmithüsen, "Veech Groups of Origamis," Dissertation 2005, Universität Karlsruhe.
[17]	G. Schmithüsen, An algorithm for finding the Veech group of an origami, Experiment. Math., 13 (2004), 459-472.
[18]	G. Schmithüsen, Examples for Veech groups of origamis, The geometry of Riemann surfaces and abelian varieties, 193-206, Contemp. Math., 397, Amer. Math. Soc., Providence, RI, 2006.
[19]	G. Schmithüsen, Origamis with non-congruence Veech groups, Proceedings of 34th Symposium on Transformation Groups, 31-55, Wing Co., Wakayama, 2007.
[20]	F. Valdez, Billiards in polygons and homogeneous foliations on $\CC^2$, Ergod. Th. & Dynam. Sys., 29 (2009), 255-271.
[21]	F. Valdez, Veech groups, irrational billiards and stable abelian differentials, Preprint 2009, arXiv:0905.1591v2.
[22]	W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.

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