Skip to content
DraconiticResearch
Search Sources API
197,785 studies31 sourcesUpdated 4 min ago
PreprintCC BY 4.0arXiv

On the kissing number of hyperbolic surfaces with a small systole

  • S. Barré
  • M. Pichot
Open at sourcePDFDOICite

Abstract

Parlier proved that there exists a constant U>0 such that for any hyperbolic surface S of genus g with systole ℓ the following holds: Kiss(S)≤ U eℓ/ℓ g where Kiss(S) is the maximum number of homotopically distinct systoles in S (Kissing number). We show that there exists a constant V>0 and a sequence Sn of hyperbolic homogeneous surfaces of genus gn→ ∞ with a bounded systole such that Kiss(Sn)≥ Vgn for all n≥ 1. This shows that Parlier's theorem is sharp for a fixed systole length. Our proof uses Bruhat--Tits buildings of type Ã2 and Hamiltonian surfaces.

Related

  1. Finite covers of a product of surfaces with bounded rank and arbitrarily large systole
    Koji Fujiwara
    arXivPreprintarXiv licenceFull textAlgebra & number theoryGeometry & topologyarXiv
  2. Improved lower bounds for the kissing number in dimensions 49 to 61
    Dmitry Kamenetsky
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsGeometry & topologyarXiv
  3. Large-Scale Autonomous Discovery of Kissing Number Constructions
    Shuxing Yang, Rui Zhao, Junyao Wu +13
    arXivPreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
  4. Lattices with Large Kissing Number from Local Density
    Huck Bennett, Minki Hhan, Zihan Li +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryGeometry & topologyarXiv
  5. 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

Cite

BibTeXRIS
BibTeX
@misc{barre2026kissing,
  title = {{On the kissing number of hyperbolic surfaces with a small systole}},
  author = {S. Barré and M. Pichot},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24222},
  eprint = {2610.24222},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24222},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the kissing number of hyperbolic surfaces with a small systole
AU  - S. Barré
AU  - M. Pichot
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.24222
UR  - https://arxiv.org/abs/2610.24222
AB  - Parlier proved that there exists a constant U>0 such that for any hyperbolic surface S of genus g with systole ℓ the following holds: Kiss(S)≤ U e^ℓ/ℓ g where Kiss(S) is the maximum number of homotopically distinct systoles in S (Kissing number). We show that there exists a constant V>0 and a sequence S_n of hyperbolic homogeneous surfaces of genus g_n→ ∞ with a bounded systole such that Kiss(S_n)≥ Vg_n for all n≥ 1. This shows that Parlier's theorem is sharp for a fixed systole length. Our proof uses Bruhat--Tits buildings of type Ã_2 and Hamiltonian surfaces.
ER  -