Teichm\"uller discs in Schottkyspace

TL;DR

This paper investigates the images of Teichmüller discs with parabolic elements in their Veech groups within Schottky space, providing constructions, algorithms, and examples for origamis and general flat surfaces.

Contribution

It demonstrates how to construct Schottky coverings that transform Teichmüller discs into punctured discs or non-discs, extending results from origamis to general flat surfaces.

Findings

Constructed Schottky coverings for origamis and flat surfaces.

Provided an algorithm to find subgroups related to origamis.

Included illustrative examples and introductory explanations.

Abstract

Take a Teichm\"uller disc whose corresponding flat surface has a Veech-Group that contains a parabolic element. We look at its image in Schottkyspace and show that we can always construct a Schottky covering such that this image is not a disc, and, in the case of translation surfaces, even a punctured disc. Firstly, we prove this for origamis (= square tiled closed translation surfaces) and, then, generalise the results to flat (resp. translation) surfaces. We also give an algorithm to find the corresponding subgroup of the fundamental group of the origami, and we give some examples. Moreover, we include an introduction to Teichm\"uller discs and Schottky space for the convenience of the reader.