# American Institute of Mathematical Sciences

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

## Infinite translation surfaces with infinitely generated Veech groups

 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.
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, (2009).   Google Scholar [2] R. G. Burns and A. M. Brunner, Two remarks on the group property of Howson,, Algebra Logic, 18 (1980), 319.  doi: 10.1007/BF01673500.  Google Scholar [3] R. Chamanara, Affine automorphism groups of surfaces of infinite type,, In the tradition of Ahlfors and Bers, (2004), 123.   Google Scholar [4] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191.  doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar [5] F. Herrlich, Teichmüller curves defined by characteristic origamis,, The geometry of Riemann surfaces and abelian varieties, (2006), 133.   Google Scholar [6] W. P. Hooper, Dynamics on an infinite surface with the lattice property,, (2007) preprint, (2007).   Google Scholar [7] W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, to appear in Annales de L'Institut Fourier (2009)., (2009).   Google Scholar [8] P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $H(2)$,, Israel J. Math., 151 (2006), 281.  doi: 10.1007/BF02777365.  Google Scholar [9] P. Hubert and S. Lelièvre, Noncongruence subgroups in $H(2)$,, Int. Math. Res. Not., (2005), 47.   Google Scholar [10] P. Hubert and T. Schmidt, Infinitely generated Veech groups,, Duke Math. J., 123 (2004), 49.  doi: 10.1215/S0012-7094-04-12312-8.  Google Scholar [11] P. Hubert and B. Weiss, Dynamics on the infinite staircase,, (2008) preprint., (2008).   Google Scholar [12] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Invent. Math., 153 (2003), 631.  doi: 10.1007/s00222-003-0303-x.  Google Scholar [13] S. Lelièvre and R. Silhol, Multi-geodesic tessellations, fractional Dehn twists and uniformization of algebraic curves,, (2007) preprint, (2007).   Google Scholar [14] C. McMullen, Teichmüller geodesics of infinite complexity,, Acta Math., 191 (2003), 191.  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] G. Schmithüsen, "Veech Groups of Origamis,", Dissertation 2005, (2005).   Google Scholar [17] G. Schmithüsen, An algorithm for finding the Veech group of an origami,, Experiment. Math., 13 (2004), 459.   Google Scholar [18] G. Schmithüsen, Examples for Veech groups of origamis,, The geometry of Riemann surfaces and abelian varieties, (2006), 193.   Google Scholar [19] G. Schmithüsen, Origamis with non-congruence Veech groups,, Proceedings of 34th Symposium on Transformation Groups, (2007), 31.   Google Scholar [20] F. Valdez, Billiards in polygons and homogeneous foliations on $\CC^2$,, Ergod. Th. & Dynam. Sys., 29 (2009), 255.   Google Scholar [21] F. Valdez, Veech groups, irrational billiards and stable abelian differentials,, Preprint 2009, (2009).   Google Scholar [22] W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards,, Invent. Math., 97 (1989), 553.  doi: 10.1007/BF01388890.  Google Scholar

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##### References:
 [1] J. Bowman, "Flat Structures and Complex Structures in Teichmüller Theory,", Thesis (Ph.D.)–Cornell University. ProQuest LLC, (2009).   Google Scholar [2] R. G. Burns and A. M. Brunner, Two remarks on the group property of Howson,, Algebra Logic, 18 (1980), 319.  doi: 10.1007/BF01673500.  Google Scholar [3] R. Chamanara, Affine automorphism groups of surfaces of infinite type,, In the tradition of Ahlfors and Bers, (2004), 123.   Google Scholar [4] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191.  doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar [5] F. Herrlich, Teichmüller curves defined by characteristic origamis,, The geometry of Riemann surfaces and abelian varieties, (2006), 133.   Google Scholar [6] W. P. Hooper, Dynamics on an infinite surface with the lattice property,, (2007) preprint, (2007).   Google Scholar [7] W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, to appear in Annales de L'Institut Fourier (2009)., (2009).   Google Scholar [8] P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $H(2)$,, Israel J. Math., 151 (2006), 281.  doi: 10.1007/BF02777365.  Google Scholar [9] P. Hubert and S. Lelièvre, Noncongruence subgroups in $H(2)$,, Int. Math. Res. Not., (2005), 47.   Google Scholar [10] P. Hubert and T. Schmidt, Infinitely generated Veech groups,, Duke Math. J., 123 (2004), 49.  doi: 10.1215/S0012-7094-04-12312-8.  Google Scholar [11] P. Hubert and B. Weiss, Dynamics on the infinite staircase,, (2008) preprint., (2008).   Google Scholar [12] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Invent. Math., 153 (2003), 631.  doi: 10.1007/s00222-003-0303-x.  Google Scholar [13] S. Lelièvre and R. Silhol, Multi-geodesic tessellations, fractional Dehn twists and uniformization of algebraic curves,, (2007) preprint, (2007).   Google Scholar [14] C. McMullen, Teichmüller geodesics of infinite complexity,, Acta Math., 191 (2003), 191.  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] G. Schmithüsen, "Veech Groups of Origamis,", Dissertation 2005, (2005).   Google Scholar [17] G. Schmithüsen, An algorithm for finding the Veech group of an origami,, Experiment. Math., 13 (2004), 459.   Google Scholar [18] G. Schmithüsen, Examples for Veech groups of origamis,, The geometry of Riemann surfaces and abelian varieties, (2006), 193.   Google Scholar [19] G. Schmithüsen, Origamis with non-congruence Veech groups,, Proceedings of 34th Symposium on Transformation Groups, (2007), 31.   Google Scholar [20] F. Valdez, Billiards in polygons and homogeneous foliations on $\CC^2$,, Ergod. Th. & Dynam. Sys., 29 (2009), 255.   Google Scholar [21] F. Valdez, Veech groups, irrational billiards and stable abelian differentials,, Preprint 2009, (2009).   Google Scholar [22] W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards,, Invent. Math., 97 (1989), 553.  doi: 10.1007/BF01388890.  Google Scholar
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