An element of PSL(2,R) lies in the Veech group of a flat surface with two finite Jenkins-Strebel directions exactly when it preserves those directions and induces an isomorphism between the associated extended origami decompositions.
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A characterization of Veech groups in terms of origamis
An element of PSL(2,R) lies in the Veech group of a flat surface with two finite Jenkins-Strebel directions exactly when it preserves those directions and induces an isomorphism between the associated extended origami decompositions.