# 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:

show all references

##### References:
 [1] Kiyoshi Igusa, Gordana Todorov. Picture groups and maximal green sequences. Electronic Research Archive, , () : -. doi: 10.3934/era.2021025 [2] Thomas Kappeler, Yannick Widmer. On nomalized differentials on spectral curves associated with the sinh-Gordon equation. Journal of Geometric Mechanics, 2021, 13 (1) : 73-143. doi: 10.3934/jgm.2020023 [3] Lei Zhang, Luming Jia. Near-field imaging for an obstacle above rough surfaces with limited aperture data. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021024 [4] Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141 [5] Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li. Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, , () : -. doi: 10.3934/era.2021022 [6] Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281 [7] Seung-Yeal Ha, Myeongju Kang, Bora Moon. Collective behaviors of a Winfree ensemble on an infinite cylinder. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2749-2779. doi: 10.3934/dcdsb.2020204 [8] Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004 [9] Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881 [10] Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066 [11] Andrés Contreras, Juan Peypouquet. Forward-backward approximation of nonlinear semigroups in finite and infinite horizon. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021051 [12] Wenjing Liu, Rong Yang, Xin-Guang Yang. Dynamics of a 3D Brinkman-Forchheimer equation with infinite delay. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021052 [13] V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066 [14] Yosra Soussi. Stable recovery of a non-compactly supported coefficient of a Schrödinger equation on an infinite waveguide. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021022 [15] Masashi Wakaiki, Hideki Sano. Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021021 [16] Yuxin Tan, Yijing Sun. The Orlicz Minkowski problem involving $0 < p < 1$: From one constant to an infinite interval. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021037

2019 Impact Factor: 0.465