Veech groups of flat surfaces with poles

Abstract

Flat surfaces that correspond to meromorphic 1-forms with poles or to meromorphic quadratic differentials containing poles of order two and higher have infinite flat area. We classify groups that appear as Veech groups of translation surfaces with poles. We characterize those surfaces such that their GL + (2,ℝ)-orbit or their SL(2,ℝ)-orbit is closed. Finally, we provide a way to determine the Veech group for a typical infinite surface in any given chamber of a stratum.