Veech groups, irrational billiards and stable abelian differentials

TL;DR

This paper investigates the Veech groups associated with flat surfaces from irrational polygonal billiards and stable abelian differentials, revealing their non-discrete nature and calculating their rank.

Contribution

It provides a detailed analysis of Veech groups for irrational billiards and stable differentials, including their non-discreteness and rank computation, advancing understanding of their geometric properties.

Findings

Veech groups are non-discrete subgroups of SO(2,R) for irrational billiards

Calculated the rank of these Veech groups

Described Veech groups for surfaces from irrational polygons and stable differentials

Abstract

We describe Veech groups of flat surfaces arising from irrational angled polygonal billiards or irreducible stable abelian differentials. For irrational polygonal billiards, we prove that these groups are non-discrete subgroups of SO(2,R) and we calculate their rank.