Realizing groups as symmetries of infinite translation surfaces

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 $\operatorname{GL}^+(2,\mathbb{R})$ 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.

Figures (7)

• Figure 1: Examples of infinite-type surfaces. Right to left: a surface with self-similar end space, a translatable surface, and a surface with a non-displaceable subsurface of finite type.
• Figure 2: The blooming Cantor tree.
• Figure 3: An example of a translatable surface $S$, with the translation $h$ and some images of a curve under its action (in blue), together with its fundamental domain, from which we obtain the surface $S'=S/\langle h \rangle$ by identifying the two dashed boundary components.
• Figure 4: A non-displaceable subsurface.
• Figure 5: Gluing along two slits with a geodesic depicted in blue.

Theorems & Definitions (74)

• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10
• Remark 1.11