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1 Introduction
 1.1 Overview
 1.2 Notations

1 Introduction

1.1 Overview

This package provides methods for calculations with certain translation surfaces called origamis. An origami (also known as square-tiled surface) is a finite covering of a torus which is ramified at most over one point. It can be described in the following way from two permutations σ_x, σ_y ∈ S_d. We take d squares Q_1, dots, Q_d and glue the lower side of Q_i to the upper side of Q_σ_y(i) and the right side of Q_i to the left side of Q_σ_x(i). We require origamis to be connected and thus the group generated by σ_x and σ_y acts transitively on {1,...,d}. In this package we identify an origami with a pair of permutations, which acts transitively on {1, dots, d} up to simultaneous conjugation. This corresponds to renumbering the squares. By choosing a certain numbering in a canonical way one can achieve a canonical representative.

We are especially interested in the so-called Veech group of an origami. This is a finite-index subgroup of SL_2(Z) which carries a lot of information about the geometric and dynamic properties of the underlying translation surface. For further information about origamis and translation surfaces in general see e.g. [Zmi11], [Sch05], [HL06], [Ran17] and [Zor06].

1.2 Notations

Since we are mainly interested in Veech groups of origamis, which are subgroups of SL_2(Z) of finite index, we fix two generators of SL_2(Z)

S = \left( {\begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array} } \right)

and

T = \left( {\begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} } \right).

Furthermore we fix the free group F generated by tildeS and tildeT. We consider the canonical epimorphism π: F-> SL_2(Z) with π(tildeS)=S and π(tildeT)=T.

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