Invariants of translation surfaces

Hubert, Pascal; Schmidt, Thomas A.

Annales de l'Institut Fourier, Tome 51 (2001) no. 2, pp. 461-495.

Nous définissons, pour une surface de translation, un invariant de revêtement affine. Cet invariant est un raffinement du groupe de Veech. Il nous permet de construire un exemple de deux surfaces de translation qui ont le même groupe de Veech et qui ne sont pas dans le même arbre de revêtements affines.

We definite invariants of translation surfaces which refine Veech groups. These aid in exact determination of Veech groups. We give examples where two surfaces of isomorphic Veech group cannot even share a common tree of balanced affine coverings. We also show that there exist translation surfaces of isomorphic Veech groups which cannot affinely cover any common surface. We also extend a result of Gutkin and Judge and thereby give the first examples of noncompact Fuchsian groups which cannot appear as Veech groups. We give an infinite family of these.

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