Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity |
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| Authors: | Eugene Gutkin |
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| Institution: | Department of Mathematics, CALTECH, Pasadena, CA 91125, USA Max-Planck-Institut for Mathematics in the Sciences, Inselstrasse 22-26, 04103 Leipzig, Germany; Institut de Mathématiques de Luminy, 163 av de Luminy, case 907, 13288 Marseille cedex 09, France; Oregon State University, Corvallis, OR 97331, USA |
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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. |
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