Pair correlation of hyperbolic lattice angles

TL;DR

This paper conjectures an explicit formula for the pair correlation of angles between geodesic rays in hyperbolic lattices and proves it for the modular group with an elliptic point.

Contribution

It introduces a conjecture for the pair correlation of hyperbolic lattice angles and proves it in a specific case involving the modular group and elliptic points.

Findings

Conjectured explicit formula for pair correlation of lattice angles.

Proof of the conjecture for PSL(2,Z) and elliptic points.

Advances understanding of geometric distributions in hyperbolic lattices.

Abstract

Let $\omega$ be a point in the upper half plane, and let $\Gamma$ be a discrete, finite covolume subgroup of $\mathrm{PSL}_2(\mathbb{R})$. We conjecture an explicit formula for the pair correlation of the angles between geodesic rays of the lattice $\Gamma\omega$, intersected with increasingly large balls centered at $\omega$. We prove this conjecture for $\Gamma=\mathrm{PSL}_2(\mathbb{Z})$ and $\omega$ an elliptic point.