Translation Surfaces With Finite Veech Groups

TL;DR

This paper demonstrates that any finite subgroup of the general linear group over the reals can be represented as the Veech group of a specially constructed translation surface, expanding understanding of symmetries in flat geometry.

Contribution

It shows that all finite subgroups of $GL_2(\mathbb{R})$ can be realized as Veech groups, providing a comprehensive realization result in the theory of translation surfaces.

Findings

Any finite subgroup of $GL_2(\mathbb{R})$ can be realized as a Veech group.

Constructs explicit examples of translation surfaces with prescribed finite Veech groups.

Enhances classification of symmetries in translation surface dynamics.

Abstract

We prove that every finite subgroup of $GL_{2}(\mathbb{R})$ can be realized as the Veech group of some translation surface.