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Horizon saddle connections and Veech groups of infinite-type dilation surfaces
Abstract
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.
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BibTeX
@misc{medrano2026horizon,
title = {{Horizon saddle connections and Veech groups of infinite-type dilation surfaces}},
author = {Oscar Rutilio Molina Medrano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31447},
eprint = {2609.31447},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31447},
}
RIS
TY - UNPB
TI - Horizon saddle connections and Veech groups of infinite-type dilation surfaces
AU - Oscar Rutilio Molina Medrano
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31447
UR - https://arxiv.org/abs/2609.31447
AB - 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.
ER -