Closed geodesics in homology classes modulo sublattices
Abstract
Let M be a Weil-Petersson random hyperbolic surface of genus g, and let Γ⊂ ℤ2g be a lattice of prime index q. We study the distribution of primitive closed geodesics in homology classes mod Γ in the large genus limit. Averaging over all lattices of index q, with q → ∞, we compute all the centered moments of the corresponding weighted counting functions, and exhibit a transition between Poisson and Gaussian regimes (depending on whether X/(qlog X), the expected number of primitive geodesics in a given homology class mod Γ, tends to λ>0 or ∞). We also study the unnormalized variance GM(X,Γ) of the counts among homology classes, and show that as X → ∞, averaged over all lattices of prime index q, it is asymptotic to Xlog X in the large genus limit. These results are analogous to phenomena arising in the distribution of primes in arithmetic progressions.
Related
- Regular Sublattices, O Closed Ideals and Dedekind MacNeille Completions
- How many points does an affine algebraic set have in residue classes modulo n ?
- Complex hyperbolic surfaces whose closed geodesics are all simple
- Closed geodesics on compact semi-Riemannian manifolds
- Growth of Simple Closed Geodesics in Finite-Volume Hyperbolic Manifolds
BibTeX
@misc{pirani2026closed,
title = {{Closed geodesics in homology classes modulo sublattices}},
author = {Noam Pirani},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.26311},
eprint = {2608.26311},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.26311},
}
RIS
TY - UNPB
TI - Closed geodesics in homology classes modulo sublattices
AU - Noam Pirani
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2608.26311
UR - https://arxiv.org/abs/2608.26311
AB - Let M be a Weil-Petersson random hyperbolic surface of genus g, and let Γ⊂ ℤ^2g be a lattice of prime index q. We study the distribution of primitive closed geodesics in homology classes mod Γ in the large genus limit. Averaging over all lattices of index q, with q → ∞, we compute all the centered moments of the corresponding weighted counting functions, and exhibit a transition between Poisson and Gaussian regimes (depending on whether X/(qlog X), the expected number of primitive geodesics in a given homology class mod Γ, tends to λ>0 or ∞). We also study the unnormalized variance G_M(X,Γ) of the counts among homology classes, and show that as X → ∞, averaged over all lattices of prime index q, it is asymptotic to Xlog X in the large genus limit. These results are analogous to phenomena arising in the distribution of primes in arithmetic progressions.
ER -