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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

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
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show all references

References:
[1]

Thesis (Ph.D.)–Cornell University. ProQuest LLC, Ann Arbor, MI, 2009.  Google Scholar

[2]

Algebra Logic, 18 (1980), 319-325. doi: 10.1007/BF01673500.  Google Scholar

[3]

In the tradition of Ahlfors and Bers, III, 123-145, Contemp. Math., 355, Amer. Math. Soc., Providence, RI, 2004.  Google Scholar

[4]

Duke Math. J., 103 (2000), 191-213. doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar

[5]

The geometry of Riemann surfaces and abelian varieties, 133-144, Contemp. Math., 397, Amer. Math. Soc., Providence, RI, 2006.  Google Scholar

[6]

(2007) preprint, arXiv:0802.0189. Google Scholar

[7]

to appear in Annales de L'Institut Fourier (2009). Google Scholar

[8]

Israel J. Math., 151 (2006), 281-321. doi: 10.1007/BF02777365.  Google Scholar

[9]

Int. Math. Res. Not., (2005), 47-64.  Google Scholar

[10]

Duke Math. J., 123 (2004), 49-69. doi: 10.1215/S0012-7094-04-12312-8.  Google Scholar

[11]

(2008) preprint. Google Scholar

[12]

Invent. Math., 153 (2003), 631-678. doi: 10.1007/s00222-003-0303-x.  Google Scholar

[13]

(2007) preprint, arXiv:math/0702374. Google Scholar

[14]

Acta Math., 191 (2003), 191-223. doi: 10.1007/BF02392964.  Google Scholar

[15]

P. Przytycki, G. Schmithüsen and F. Valdez, Veech groups of Loch Ness monsters,, to appear in Annales de l'Institut Fourier., ().   Google Scholar

[16]

Dissertation 2005, Universität Karlsruhe. Google Scholar

[17]

Experiment. Math., 13 (2004), 459-472.  Google Scholar

[18]

The geometry of Riemann surfaces and abelian varieties, 193-206, Contemp. Math., 397, Amer. Math. Soc., Providence, RI, 2006.  Google Scholar

[19]

Proceedings of 34th Symposium on Transformation Groups, 31-55, Wing Co., Wakayama, 2007.  Google Scholar

[20]

Ergod. Th. & Dynam. Sys., 29 (2009), 255-271.  Google Scholar

[21]

Preprint 2009, arXiv:0905.1591v2. Google Scholar

[22]

Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.  Google Scholar

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