Veech groups of infinite-genus surfaces

Abstract

We show that every countable subgroup $G<{GL}_{+}\left(2,ℝ\right)$ 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 realize all possible uncountable Veech groups.