Veech groups of infinite genus surfaces

TL;DR

This paper demonstrates that any countable subgroup of GL_+(2,R) without contracting elements can be realized as the Veech group of a tame infinite genus translation surface, with no topological restrictions if all ends have genus.

Contribution

It constructs infinite genus translation surfaces with prescribed Veech groups, extending the understanding of Veech groups beyond finite genus cases.

Findings

Any countable subgroup without contracting elements is realizable as a Veech group.

No topological restrictions on the surface if every end has genus.

Infinite genus surfaces can realize all uncountable Veech groups.

Abstract

We show that every countable subgroup $G<\rm GL_+(2,\mathbb{R})$ without contracting elements is the Veech group of a tame translation surface $S$ of infinite genus, for infinitely many different topological types of $S$. Moreover, we prove that as long as every end has genus, there are no restrictions on the topological type of $S$ to realise all possible uncountable Veech groups.