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 \(\sigma_x, \sigma_y \in S_d\). We take \(d\) squares \(Q_1, \dots, Q_d\) and glue the lower side of \(Q_i\) to the upper side of \(Q_{\sigma_y(i)}\) and the right side of \(Q_i\) to the left side of \(Q_{\sigma_x(i)}\). We require origamis to be connected and thus the group generated by \(\sigma_x\) and \(\sigma_y\) acts transitively on \(\{1,\ldots,d\}\). 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 \(SL_2(\mathbb{Z})\) 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].
Since we are mainly interested in Veech groups of origamis, which are subgroups of \(SL_2(\mathbb{Z})\) of finite index, we fix two generators of \(SL_2(\mathbb{Z})\)
\[ S = \left( {\begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array} } \right) \]
and
\[ T = \left( {\begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} } \right). \]
Furthermore we fix the free group \(F\) generated by \(\tilde{S}\) and \(\tilde{T}\). We consider the canonical epimorphism \(\pi: F\to SL_2(\mathbb{Z})\) with \(\pi(\tilde{S})=S\) and \(\pi(\tilde{T})=T\).
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