Limit theorems for cuspidal orbits on Fuchsian groups
Abstract
Let Γ⊂ SL(2, ℝ) be discrete, co-finite with cusps, and let frak b∈ ℝ∪{∞} be a cusp of Γ. Let f: Γfrak b → ℝ be a quantum modular form in the sense that the maps ϕγ(x) = f(x) − f(γx) (x∈ Γfrak b, γ∈ Γ) extend to Hölder-regular functions. Under bounded moments assumptions of the shape ∑n n−2 ∫ |ϕγn(x)|αd x < ∞ for some α>0 and every fixed parabolic element γ∈ Γ, we prove asymptotic formulas for the characteristic functions of the values f(x) for x∈ Γfrak b, ordered by growing denominators. These yield Gaussian limits when α>2 and other stable limit laws under suitable further assumptions. This generalizes to Fuchsian groups the work of Baladi-Vallée on rational orbits of the Gauss map. In companion papers, we use this result, and in particular its flexible assumptions on ϕγ, to deduce limit theorems for automorphic L-series at the central point. One ingredient in our proof is a particular cuspidal acceleration of the Bowen-Series map, whose inverse branches admit a convenient parametrization, and for which we describe explicitly a natural extension.
Related
- Quasi-Fuchsian groups and complex realisations of q-deformed real numbers
- Medvedev degrees of SFTs on cocompact Fuchsian groups
- On the II1 Factors of Fuchsian Groups
- Vertex operator algebra bundles on Riemann surfaces of higher genus and automorphic forms for Fuchsian groups
- Tate cohomology of unipotent cuspidal representations of finite groups of Lie type
BibTeX
@misc{bettin2026limit,
title = {{Limit theorems for cuspidal orbits on Fuchsian groups}},
author = {Sandro Bettin and Sary Drappeau and Jungwon Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41973},
eprint = {2610.41973},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41973},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Limit theorems for cuspidal orbits on Fuchsian groups
AU - Sandro Bettin
AU - Sary Drappeau
AU - Jungwon Lee
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41973
UR - https://arxiv.org/abs/2610.41973
AB - Let Γ⊂ SL(2, ℝ) be discrete, co-finite with cusps, and let frak b∈ ℝ∪{∞} be a cusp of Γ. Let f: Γfrak b → ℝ be a quantum modular form in the sense that the maps ϕ_γ(x) = f(x) − f(γx) (x∈ Γfrak b, γ∈ Γ) extend to Hölder-regular functions. Under bounded moments assumptions of the shape ∑_n n^−2 ∫ |ϕ_γ^n(x)|^αd x < ∞ for some α>0 and every fixed parabolic element γ∈ Γ, we prove asymptotic formulas for the characteristic functions of the values f(x) for x∈ Γfrak b, ordered by growing denominators. These yield Gaussian limits when α>2 and other stable limit laws under suitable further assumptions. This generalizes to Fuchsian groups the work of Baladi-Vallée on rational orbits of the Gauss map. In companion papers, we use this result, and in particular its flexible assumptions on ϕ_γ, to deduce limit theorems for automorphic L-series at the central point. One ingredient in our proof is a particular cuspidal acceleration of the Bowen-Series map, whose inverse branches admit a convenient parametrization, and for which we describe explicitly a natural extension.
ER -