Infinite translation surfaces with infinitely generated Veech groups

Previous Article
Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory
JMD Home
This Issue
Next Article
Survival of infinitely many critical points for the Rabinowitz action functional

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
Related Papers

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, (2009).
[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.
[3]	R. Chamanara, Affine automorphism groups of surfaces of infinite type,, In the tradition of Ahlfors and Bers, (2004), 123.
[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.
[5]	F. Herrlich, Teichmüller curves defined by characteristic origamis,, The geometry of Riemann surfaces and abelian varieties, (2006), 133.
[6]	W. P. Hooper, Dynamics on an infinite surface with the lattice property,, (2007) preprint, (2007).
[7]	W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, to appear in Annales de L'Institut Fourier (2009)., (2009).
[8]	P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $H(2)$,, Israel J. Math., 151 (2006), 281. doi: 10.1007/BF02777365.
[9]	P. Hubert and S. Lelièvre, Noncongruence subgroups in $H(2)$,, Int. Math. Res. Not., (2005), 47.
[10]	P. Hubert and T. Schmidt, Infinitely generated Veech groups,, Duke Math. J., 123 (2004), 49. doi: 10.1215/S0012-7094-04-12312-8.
[11]	P. Hubert and B. Weiss, Dynamics on the infinite staircase,, (2008) preprint., (2008).
[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.
[13]	S. Lelièvre and R. Silhol, Multi-geodesic tessellations, fractional Dehn twists and uniformization of algebraic curves,, (2007) preprint, (2007).
[14]	C. McMullen, Teichmüller geodesics of infinite complexity,, Acta Math., 191 (2003), 191. 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, (2005).
[17]	G. Schmithüsen, An algorithm for finding the Veech group of an origami,, Experiment. Math., 13 (2004), 459.
[18]	G. Schmithüsen, Examples for Veech groups of origamis,, The geometry of Riemann surfaces and abelian varieties, (2006), 193.
[19]	G. Schmithüsen, Origamis with non-congruence Veech groups,, Proceedings of 34th Symposium on Transformation Groups, (2007), 31.
[20]	F. Valdez, Billiards in polygons and homogeneous foliations on $\CC^2$,, Ergod. Th. & Dynam. Sys., 29 (2009), 255.
[21]	F. Valdez, Veech groups, irrational billiards and stable abelian differentials,, Preprint 2009, (2009).
[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.

show all references

References:

[1]	J. Bowman, "Flat Structures and Complex Structures in Teichmüller Theory,", Thesis (Ph.D.)–Cornell University. ProQuest LLC, (2009).
[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.
[3]	R. Chamanara, Affine automorphism groups of surfaces of infinite type,, In the tradition of Ahlfors and Bers, (2004), 123.
[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.
[5]	F. Herrlich, Teichmüller curves defined by characteristic origamis,, The geometry of Riemann surfaces and abelian varieties, (2006), 133.
[6]	W. P. Hooper, Dynamics on an infinite surface with the lattice property,, (2007) preprint, (2007).
[7]	W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, to appear in Annales de L'Institut Fourier (2009)., (2009).
[8]	P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $H(2)$,, Israel J. Math., 151 (2006), 281. doi: 10.1007/BF02777365.
[9]	P. Hubert and S. Lelièvre, Noncongruence subgroups in $H(2)$,, Int. Math. Res. Not., (2005), 47.
[10]	P. Hubert and T. Schmidt, Infinitely generated Veech groups,, Duke Math. J., 123 (2004), 49. doi: 10.1215/S0012-7094-04-12312-8.
[11]	P. Hubert and B. Weiss, Dynamics on the infinite staircase,, (2008) preprint., (2008).
[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.
[13]	S. Lelièvre and R. Silhol, Multi-geodesic tessellations, fractional Dehn twists and uniformization of algebraic curves,, (2007) preprint, (2007).
[14]	C. McMullen, Teichmüller geodesics of infinite complexity,, Acta Math., 191 (2003), 191. 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, (2005).
[17]	G. Schmithüsen, An algorithm for finding the Veech group of an origami,, Experiment. Math., 13 (2004), 459.
[18]	G. Schmithüsen, Examples for Veech groups of origamis,, The geometry of Riemann surfaces and abelian varieties, (2006), 193.
[19]	G. Schmithüsen, Origamis with non-congruence Veech groups,, Proceedings of 34th Symposium on Transformation Groups, (2007), 31.
[20]	F. Valdez, Billiards in polygons and homogeneous foliations on $\CC^2$,, Ergod. Th. & Dynam. Sys., 29 (2009), 255.
[21]	F. Valdez, Veech groups, irrational billiards and stable abelian differentials,, Preprint 2009, (2009).
[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.

[1]	Eduard Duryev, Charles Fougeron, Selim Ghazouani. Dilation surfaces and their Veech groups. Journal of Modern Dynamics, 2019, 14: 121-151. doi: 10.3934/jmd.2019005
[2]	Ferrán Valdez. Veech groups, irrational billiards and stable abelian differentials. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1055-1063. doi: 10.3934/dcds.2012.32.1055
[3]	Artur Avila, Carlos Matheus, Jean-Christophe Yoccoz. The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces. Journal of Modern Dynamics, 2019, 14: 21-54. doi: 10.3934/jmd.2019002
[4]	David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477
[5]	Kariane Calta, John Smillie. Algebraically periodic translation surfaces. Journal of Modern Dynamics, 2008, 2 (2) : 209-248. doi: 10.3934/jmd.2008.2.209
[6]	Benjamin Dozier. Equidistribution of saddle connections on translation surfaces. Journal of Modern Dynamics, 2019, 14: 87-120. doi: 10.3934/jmd.2019004
[7]	Pierre Arnoux, Thomas A. Schmidt. Veech surfaces with nonperiodic directions in the trace field. Journal of Modern Dynamics, 2009, 3 (4) : 611-629. doi: 10.3934/jmd.2009.3.611
[8]	Alexander I. Bufetov. Hölder cocycles and ergodic integrals for translation flows on flat surfaces. Electronic Research Announcements, 2010, 17: 34-42. doi: 10.3934/era.2010.17.34
[9]	Eugene Gutkin. Insecure configurations in lattice translation surfaces, with applications to polygonal billiards. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 367-382. doi: 10.3934/dcds.2006.16.367
[10]	Nancy Guelman, Isabelle Liousse. Actions of Baumslag-Solitar groups on surfaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1945-1964. doi: 10.3934/dcds.2013.33.1945
[11]	Juan Alonso, Nancy Guelman, Juliana Xavier. Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo)-Anosov elements. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1817-1827. doi: 10.3934/dcds.2015.35.1817
[12]	Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481
[13]	Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1333-1349. doi: 10.3934/dcds.2013.33.1333
[14]	Percy Fernández-Sánchez, Jorge Mozo-Fernández, Hernán Neciosup. Dicritical nilpotent holomorphic foliations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3223-3237. doi: 10.3934/dcds.2018140
[15]	Dae San Kim. Infinite families of recursive formulas generating power moments of ternary Kloosterman sums with square arguments arising from symplectic groups. Advances in Mathematics of Communications, 2009, 3 (2) : 167-178. doi: 10.3934/amc.2009.3.167
[16]	The Editors. William A. Veech's publications. Journal of Modern Dynamics, 2019, 14: ⅰ-ⅳ. doi: 10.3934/jmd.2019i
[17]	José A. Conejero, Alfredo Peris. Chaotic translation semigroups. Conference Publications, 2007, 2007 (Special) : 269-276. doi: 10.3934/proc.2007.2007.269
[18]	John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047
[19]	Toshikazu Ito, Bruno Scárdua. Holomorphic foliations transverse to manifolds with corners. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 537-544. doi: 10.3934/dcds.2009.25.537
[20]	Dawei Chen. Strata of abelian differentials and the Teichmüller dynamics. Journal of Modern Dynamics, 2013, 7 (1) : 135-152. doi: 10.3934/jmd.2013.7.135

2017 Impact Factor: 0.425

American Institute of Mathematical Sciences