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Simple geodesics in closed hyperbolic manifolds
Abstract
Given any closed hyperbolic n-manifold M (n≥ 3), we show that for a fixed κ>3/(n−2), a random closed geodesic of length close to r has the collar of width r−κ when r is large enough. In particular, most closed geodesics of length close to r are simple. This theorem positively answers Problem 3.16 from Kirby's list which asks whether every closed hyperbolic 3-manifold contains infinitely many simple closed geodesics.
Related
- Growth of Simple Closed Geodesics in Finite-Volume Hyperbolic Manifolds
- Complex hyperbolic surfaces whose closed geodesics are all simple
- Closed geodesics on compact semi-Riemannian manifolds
- Sharp effective equidistribution of closed geodesics in compact hyperbolic surfaces
- Lengths of closed geodesics on random non-compact hyperbolic surfaces
BibTeX
@misc{luo2026simple,
title = {{Simple geodesics in closed hyperbolic manifolds}},
author = {Qiliang Luo and Vladimir Marković},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29761},
eprint = {2608.29761},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29761},
}
RIS
TY - UNPB
TI - Simple geodesics in closed hyperbolic manifolds
AU - Qiliang Luo
AU - Vladimir Marković
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29761
UR - https://arxiv.org/abs/2608.29761
AB - Given any closed hyperbolic n-manifold M (n≥ 3), we show that for a fixed κ>3/(n−2), a random closed geodesic of length close to r has the collar of width r^−κ when r is large enough. In particular, most closed geodesics of length close to r are simple. This theorem positively answers Problem 3.16 from Kirby's list which asks whether every closed hyperbolic 3-manifold contains infinitely many simple closed geodesics.
ER -