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Equidistribution of partial coverings defined from closed geodesics

Asbjørn Christian Nordentoft, Ser Peow Tan

TL;DR

The paper proves that equidistribution of oriented closed geodesics in the unit tangent bundle implies equidistribution of the associated partial coverings on a finite-volume hyperbolic surface. It introduces a purely geometric construction of partial coverings via painting to the left of geodesics relative to a fundamental polygon and leverages Stokes’ theorem, volume lower bounds, and boundary/topology arguments to transfer equidistribution from geodesics to coverings. The approach provides a theta-free route to Duke–Imamoḡlu–Tóth-type results for the modular surface and yields effective statements in the cocompact case while addressing cusp phenomena for non-compact quotients through non-escape-of-mass techniques. Applications include broad equidistribution results for not-too-thin geodesic families, level-$M$ modular settings, sparse class-group–driven families, and $q$-orbit constructions, thereby unifying and extending several previous results in the literature. Overall, the work connects purely geometric combinatorics of polygons and geodesics with automorphic-analytic tools to obtain new distributional results for partial coverings on hyperbolic surfaces.

Abstract

Given a finite volume hyperbolic surface, a fundamental polygon and an oriented closed geodesic, we associate a partial covering of the surface. We prove that given a sequence of collections of oriented closed geodesics equidistributing in the unit tangent bundle then the associated partial coverings also equidistribute. In the case of the modular group, this yields an alternative proof of an equidistribution result due to Duke, Imamoglu, and Tóth.

Equidistribution of partial coverings defined from closed geodesics

TL;DR

The paper proves that equidistribution of oriented closed geodesics in the unit tangent bundle implies equidistribution of the associated partial coverings on a finite-volume hyperbolic surface. It introduces a purely geometric construction of partial coverings via painting to the left of geodesics relative to a fundamental polygon and leverages Stokes’ theorem, volume lower bounds, and boundary/topology arguments to transfer equidistribution from geodesics to coverings. The approach provides a theta-free route to Duke–Imamoḡlu–Tóth-type results for the modular surface and yields effective statements in the cocompact case while addressing cusp phenomena for non-compact quotients through non-escape-of-mass techniques. Applications include broad equidistribution results for not-too-thin geodesic families, level- modular settings, sparse class-group–driven families, and -orbit constructions, thereby unifying and extending several previous results in the literature. Overall, the work connects purely geometric combinatorics of polygons and geodesics with automorphic-analytic tools to obtain new distributional results for partial coverings on hyperbolic surfaces.

Abstract

Given a finite volume hyperbolic surface, a fundamental polygon and an oriented closed geodesic, we associate a partial covering of the surface. We prove that given a sequence of collections of oriented closed geodesics equidistributing in the unit tangent bundle then the associated partial coverings also equidistribute. In the case of the modular group, this yields an alternative proof of an equidistribution result due to Duke, Imamoglu, and Tóth.

Paper Structure

This paper contains 24 sections, 27 theorems, 152 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Applications
  4. Idea of proof
  5. Background on automorphic forms
  6. Cofinite hyperbolic orbifolds
  7. Automorphic forms
  8. Some spectral theory (compact case)
  9. Incomplete Poincaré series
  10. Topology of fundamental polygons
  11. Fundamental polygons
  12. Structure of the boundary of fundamental polygons
  13. Partial coverings defined from geodesics (''painting to the left'')
  14. Comparing with the construction of Duke--Imamoḡlu--Tóth
  15. Comparing with the construction of Humphries--Nordentoft
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\mathcal{F}$ be a fundamental polygon for a discrete and cofinite subgroup $\Gamma\leq \mathrm{PSL}_2(\mathbb{R})$. Let $\mathbf{C}_1,\mathbf{C}_2,\ldots$ be a sequence of collections of oriented closed geodesics on $Y_\Gamma$ equidistributing in the unit tangent bundle $\Gamma\backslash \mathr

Figures (3)

  • Figure 1: Polygons $\mathcal{P}_1,\ldots, \mathcal{P}_m$ (shaded) obtained from the closed geodesics $\mathcal{C}_1$ and $\mathcal{C}_2$ when $\Gamma=\Gamma_\triangle$ and $\mathcal{F}=\mathcal{F}_\triangle$.
  • Figure 2: Polygons (shaded) obtained from the closed geodesic $\mathcal{C}_3$ when $\Gamma=\Gamma_\triangle$ and $\mathcal{F}=\mathcal{F}_\triangle$.
  • Figure 3: Partial covering obtained from the pair $(\mathcal{C}_3,\mathcal{F}_\triangle)$.

Theorems & Definitions (62)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 52 more