# Origamis with non congruence Veech groups

## TL;DR

This paper demonstrates that for any genus greater than one, there exist translation surfaces with Veech groups that are non-congruence subgroups of SL(2,Z), constructed using origamis and square-tiled surfaces.

## Contribution

It introduces a method to construct translation surfaces with non-congruence Veech groups for all genera greater than one, expanding understanding of Veech group properties.

## Key findings

- Existence of translation surfaces with non-congruence Veech groups for all g > 1
- Explicit examples of genus 2 origamis with non-congruence Veech groups
- A technique to generate sequences of origamis with decreasing Veech groups

## Abstract

As main result we show that for each g > 1 there is some translation surface of genus g whose Veech group is a non congruence subgroup of SL(2,Z). We use origamis/square-tiled surfaces to produce our examples. The article is divided into two parts: In the first part we introduce translation surfaces, origamis, Veech groups and Teichmueller curves and show for two origamis in genus 2 that their Veech groups are non congruence groups; in the second part we provide a technique that produces sequences of origamis whose Veech groups are decreasing. This is used to prove the main result.