Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 32 min ago
PreprintarXiv licencearXiv

Growth of Simple Closed Geodesics in Finite-Volume Hyperbolic Manifolds

  • Xiaolong Hans Han
  • Bohan Yang
Open at sourcePDFDOICite

Abstract

Let M be a finite-volume hyperbolic d-manifold with d≥3. We prove that the number of primitive nonsimple closed geodesics has exponential growth rate strictly smaller than d−1. Consequently, asymptotically almost every primitive closed geodesic in M is simple. In contrast, we show that on an arithmetic hyperbolic manifold of type~I, the unit vectors tangent to nonsimple closed geodesics are dense in the unit tangent bundle.

Related

  1. Simple geodesics in closed hyperbolic manifolds
    Qiliang Luo, Vladimir Marković
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  2. Complex hyperbolic surfaces whose closed geodesics are all simple
    Sami Douba, D. B. McReynolds, Julien Paupert +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryGeometry & topologyarXiv
  3. Closed geodesics on compact semi-Riemannian manifolds
    Souheib Allout, Salah Chaib
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  4. Sharp effective equidistribution of closed geodesics in compact hyperbolic surfaces
    Junehyuk Jung, Insung Park, Peter Zenz
    arXivPreprintarXiv licenceFull textAnalysis & PDEsDynamical systemsarXiv
  5. Lengths of closed geodesics on random non-compact hyperbolic surfaces
    Malo Gatignol, Joe Thomas
    arXivPreprintCC BY-NC-ND 4.0Full textAnalysis & PDEsGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{han2026growth,
  title = {{Growth of Simple Closed Geodesics in Finite-Volume Hyperbolic Manifolds}},
  author = {Xiaolong Hans Han and Bohan Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.29855},
  eprint = {2608.29855},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29855},
}
RIS
TY  - UNPB
TI  - Growth of Simple Closed Geodesics in Finite-Volume Hyperbolic Manifolds
AU  - Xiaolong Hans Han
AU  - Bohan Yang
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2608.29855
UR  - https://arxiv.org/abs/2608.29855
AB  - Let M be a finite-volume hyperbolic d-manifold with d≥3. We prove that the number of primitive nonsimple closed geodesics has exponential growth rate strictly smaller than d−1. Consequently, asymptotically almost every primitive closed geodesic in M is simple. In contrast, we show that on an arithmetic hyperbolic manifold of type~I, the unit vectors tangent to nonsimple closed geodesics are dense in the unit tangent bundle.
ER  -