Veech groups of infinite genus surfaces

Abstract

We show that every countable subgroup $G<\mathrm{G}{\mathrm{L}}_{+}(2,\mathbb{R})$ 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.

26 pages, 12 figures