Veech groups of flat surfaces with poles

Guillaume Tahar

Published 2016-06-12Version 1

Flat surfaces that correspond to meromorphic $1$-forms or meromorphic quadratic differentials with poles of order no smaller than two are surfaces of infinite area. Therefore, we cannot normalize the area and have to consider the whole action of $GL^{+}(2,\mathbb{R})$ and not just that of $SL(2,\mathbb{R})$ like in the case of classical flat surfaces. We classify groups that appear as Veech groups of translation surfaces with poles. We characterize those surfaces such that their $GL^{+}(2,\mathbb{R})$-orbit or their $SL(2,\mathbb{R})$-orbit is closed. Finally, we provide a way to determine the Veech group for a typical infinite surface in a given chamber of a stratum.