On the Ends of groups and the Veech groups of 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 $G<{\mathrm{GL}}_{+}(2,\mathbb{R})$ 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 $\mathcal{B}\bigsqcup \mathcal{U}$, 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 $S$.

arXiv admin note: text overlap with arXiv:1603.00503

References in corpus (5)

• Non-ergodic Z-periodic billiards and infinite translation surfaces
• Veech groups of Loch Ness monsters
• Wild translation surfaces and infinite genus
• Loxodromic elements in big mapping class groups via the Hooper-Thurston-Veech construction
• Realizing groups as symmetries of infinite translation surfaces