# On the Ends of groups and the Veech groups of infinite-genus surfaces

## TL;DR

This paper develops a method to construct infinite-genus translation surfaces with prescribed Veech groups, linking the geometric properties of the surface to the algebraic properties of the group.

## Contribution

It introduces a modified PSV construction that produces tame translation surfaces with specified finitely generated Veech groups and describes their ends space structure.

## Key findings

- Constructed surfaces have Veech groups matching given subgroups without contracting elements.
- The ends space of the surfaces decomposes into parts related to the group and a dense open subset.
- The method generalizes previous constructions to infinite-genus surfaces.

## Abstract

In this paper, we study the PSV construction, which provides a step by step method for obtaining tame translation surfaces with a suitable Veech group. In addition, we modify slightly this construction, and for each finitely generated subgroup $G<{\rm GL}_{+}(2,\mathbb{R})$ without contracting elements, we produce a tame translation surface $S$ with infinite genus such that its Veech group is $G$. Furthermore, the ends space of $S$ can be written as $\mathcal{B}\sqcup \mathcal{U}$, where $\mathcal{B}$ is homeomorphic to the ends space of the group $G$, and $\mathcal{U}$ is a countable, discrete, dense, and open subset of the ends space of $S$.