Realizing groups as symmetries of infinite translation surfaces

TL;DR

This paper classifies groups that can be isometry groups of certain infinite translation surfaces and shows how many subgroups of GL(2,R) can be realized as Veech groups under specific topological conditions.

Contribution

It provides a complete classification of isometry groups for a class of infinite translation surfaces and generalizes the realization of Veech groups to broader topological settings.

Findings

Classified isometry groups for infinite translation surfaces with non-finitely generated fundamental groups.

Every countable subgroup of GL(2,R) can be realized as a Veech group under certain conditions.

Extended previous results on Veech groups to more general infinite 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-Schmith\"{u}sen 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.