Counting Flat Surfaces with many Symmetries

A flat surface is a polygon in the plane where we have identified parallel sides or equivalently a complex curve together with a choice of a holomorphic differential form. Flat surfaces with many real symmetries give rise to special curves inside the moduli space of curves called Teichmüller curves. Flat surfaces with extra holomorphic symmetries correspond to orbifold points on such Teichmüller curves. We provide a topological classification of some Teichmüller curves by explicitly counting their orbifold points.