Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity

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Abstract

We study translation surfaces with rich groups of affine diffeomorphisms—“prelattice” surfaces. These include the lattice translation surfaces studied by W. Veech. We show that there exist prelattice but nonlattice translation surfaces. We characterize arithmetic surfaces among prelattice surfaces by the infinite cardinality of their set of points periodic under affine diffeomorphisms. We give examples of translation surfaces whose periodic points and Weierstrass points coincide.

Résumé

Nous étudions des surfaces de translation ayant un grand groupe de difféomorphisms affines—les surfaces « préréseaux ». Parmi celles-ci se trouvent les surfaces de translation réseaux étudiées par W. Veech. Nous montrons qu'il existe des surfaces de translation préréseaux qui ne sont pas réseaux. Nous donnons une nouvelle caractérisation des surfaces arithmétiques : ce sont les surfaces préréseaux qui ont un nombre infini de points périodiques sous l'action du groupe des difféomorphisms affines. Nous exhibons des exemples de surfaces de translation dont les points périodiques et points de Weierstrass coı̈ncident.

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