Horizon saddle connections and Veech groups of infinite-type dilation surfaces

This paper studies Veech groups of dilation surfaces whose fundamental group is not finitely generated. We prove that if $S$ is a surface with self-similar end space and every end is accumulated by genus, then every countable subgroup of $SL(2,\mathbb{R})$ can be realized as the Veech group of a dilation surface homeomorphic to $S$. Additionally, we can construct this dilation surface such that it realizes horizon saddle connections in a prescribed set of directions. This contrasts with the case of closed dilation surfaces, where horizon saddle connections impose restrictions on the algebraic structure of the Veech group.