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Finding the Birman-Series set
Abstract
By a celebrated result of Birman and Series, the set of simple complete geodesics form a nowhere dense subset of a hyperbolic surface. We show that a segment of finite length is in a (small) neighborhood of this set if it can be "extended simply" in both directions for long enough. We deduce that, for large enough genus g, there exist surfaces with a Birman-Series set that is roughly 1/gsfrac23-dense.
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BibTeX
@misc{buser2026finding,
title = {{Finding the Birman-Series set}},
author = {Peter Buser and Hugo Parlier},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13645},
eprint = {2610.13645},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13645},
}
RIS
TY - UNPB
TI - Finding the Birman-Series set
AU - Peter Buser
AU - Hugo Parlier
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13645
UR - https://arxiv.org/abs/2610.13645
AB - By a celebrated result of Birman and Series, the set of simple complete geodesics form a nowhere dense subset of a hyperbolic surface. We show that a segment of finite length is in a (small) neighborhood of this set if it can be "extended simply" in both directions for long enough. We deduce that, for large enough genus g, there exist surfaces with a Birman-Series set that is roughly 1/g^sfrac23-dense.
ER -