Infinite translation surfaces with infinitely generated Veech groups
Pascal Hubert Gabriela Schmithüsen
Journal of Modern Dynamics 2010, 4(4): 715-732 doi: 10.3934/jmd.2010.4.715
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.

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