Infinite translation surfaces with infinitely generated Veech groups

Abstract

We study infinite translation surfaces which are $\ZZ$-covers of finitesquare-tiled surfaces obtained by a certain two-slit cut and pasteconstruction. We show that if the finite translation surface has aone-cylinder decomposition in some direction, then the Veech group of theinfinite translation surface is either a lattice or an infinitely generatedgroup of the first kind. The square-tiled surfaces of genus two with one zeroprovide examples for finite translation surfaces that fulfill the prerequisitesof the theorem.