Veech groups of flat surfaces with poles

TL;DR

This paper classifies Veech groups of infinite-area flat surfaces with poles, characterizes surfaces with closed orbits under group actions, and provides methods to determine Veech groups for typical infinite surfaces.

Contribution

It introduces a classification of Veech groups for flat surfaces with poles and characterizes surfaces with closed orbits, advancing understanding of infinite-area translation surfaces.

Findings

Classified Veech groups for surfaces with poles.

Characterized surfaces with closed $GL^{+}(2, eal)$- or $SL(2, eal)$-orbits.

Provided a method to determine Veech groups in specific strata.

Abstract

Flat surfaces that correspond to meromorphic $1$-forms or to meromorphic quadratic differentials containing poles of order two and higher are surfaces of infinite area. 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.