Journal of Modern Dynamics (JMD)

Infinite translation surfaces with infinitely generated Veech groups

Pages: 715 - 732, Issue 4, October 2010      doi:10.3934/jmd.2010.4.715

       Abstract        References        Full Text (249.0K)       Related Articles

Pascal Hubert - LATP, case cour A, Faculté des sciences Saint Jérôme, Avenue Escadrille Normandie Niemen, 13397 Marseille cedex 20, France (email)
Gabriela Schmithüsen - Institute for Algebra and Geometry, University of Karlsruhe, 76128 Karlsruhe, Germany (email)

Abstract: 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, 37D50.

Received: June 2010;      Revised: September 2010;      Available Online: January 2011.