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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.00158 |
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| _version_ | 1866914344864841728 |
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| author | Artigiani, Mauro Randecker, Anja Sadanand, Chandrika Valdez, Ferrán Weitze-Schmithüsen, Gabriela |
| author_facet | Artigiani, Mauro Randecker, Anja Sadanand, Chandrika Valdez, Ferrán Weitze-Schmithüsen, Gabriela |
| contents | We provide a complete classification of groups that can be realized as isometry groups of a translation surface $M$ with non-finitely generated fundamental group and no planar ends. Furthermore, we demonstrate that if $S$ has no non-displaceable subsurfaces and its space of ends is self-similar, then every countable subgroup of $\operatorname{GL}^+(2,\mathbb{R})$ can be realized as the Veech group of a translation surface $M$ homeomorphic to $S$. The latter result generalizes and improves upon the previous findings of Przytycki-Valdez-Weitze-Schmithüsen and Maluendas-Valdez. To prove these results, we adapt ideas from the work of Aougab-Patel-Vlamis, which focused on hyperbolic surfaces, to translation surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_00158 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Realizing groups as symmetries of infinite translation surfaces Artigiani, Mauro Randecker, Anja Sadanand, Chandrika Valdez, Ferrán Weitze-Schmithüsen, Gabriela Geometric Topology Group Theory 57M60 (Primary) 30F99 (Secondary) We provide a complete classification of groups that can be realized as isometry groups of a translation surface $M$ with non-finitely generated fundamental group and no planar ends. Furthermore, we demonstrate that if $S$ has no non-displaceable subsurfaces and its space of ends is self-similar, then every countable subgroup of $\operatorname{GL}^+(2,\mathbb{R})$ can be realized as the Veech group of a translation surface $M$ homeomorphic to $S$. The latter result generalizes and improves upon the previous findings of Przytycki-Valdez-Weitze-Schmithüsen and Maluendas-Valdez. To prove these results, we adapt ideas from the work of Aougab-Patel-Vlamis, which focused on hyperbolic surfaces, to translation surfaces. |
| title | Realizing groups as symmetries of infinite translation surfaces |
| topic | Geometric Topology Group Theory 57M60 (Primary) 30F99 (Secondary) |
| url | https://arxiv.org/abs/2311.00158 |