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 upper side of Q_i to the lower 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].
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)
[ 0 -1 ] [ 1 1 ]
S = [ 1 0 ], T = [ 0 1 ]
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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