Generalized Staircases: Recurrence and Symmetry

Hooper, W. Patrick; Weiss, Barak

Annales de l'Institut Fourier, Tome 62 (2012) no. 4, pp. 1581-1600.

Nous étudions les $\mathbb{Z}$-revêtements de translation des surfaces de translation compactes. Nous donnons des conditions nécessaires pour que le groupe de Veech soit fuchsien du premier type, et une condition nécessaire et suffisante pour la récurrence du flot directionnel. En étendant des résultats de Hubert et Schmithüsen, nous donnons des exemples non-arithmétiques dont le groupe de Veech est un réseau et des exemples à groupe de Veech de type infini.

We study infinite translation surfaces which are $\mathbb{Z}$-covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition for recurrence of their straight-line flows. Extending results of Hubert and Schmithüsen, we provide examples of infinite non-arithmetic lattice surfaces, as well as surfaces with infinitely generated Veech groups.

Classification : 11Y40, 12Y05, 37M99, 52C99
Keywords: Infinite translation surfaces, Veech groups, lattices, straightline flow
Mot clés : Surfaces de translation infini, Groupes de Veech, Reseau, Flot directionnel

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