Infinite translation surfaces with infinitely generated Veech groups

Pascal Hubert; Gabriela Schmithüsen

doi:10.3934/jmd.2010.4.715

Article Contents

2010, Volume 4, Issue 4: 715-732. Doi: 10.3934/jmd.2010.4.715

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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
2.
Institute for Algebra and Geometry, University of Karlsruhe, 76128 Karlsruhe

Received: May 31, 2010

Revised: August 31, 2010

Published: December 2010

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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.

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Mathematics Subject Classification: 30F35, 30F30, 14H30, 37D50.

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