Realizing groups as symmetries of infinite translation surfaces

2023/10/31 by Mauro Artigiani, Artigiani, Mauro, Anja Randecker +7
Mathematics · #57M60 (Primary) 30F99 (Secondary) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2311.00158

openalex publication_date 2023/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 GL⁺(2,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.

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• On the Ends of groups and the Veech groups of infinite-genus surfaces
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