Pair correlation of hyperbolic lattice angles

Let $ω$ be a point in the upper half plane, and let $Γ$ 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 $Γω$, intersected with increasingly large balls centered at $ω$. We prove this conjecture for $Γ=\mathrm{PSL}_2(\mathbb{Z})$ and $ω$ an elliptic point.

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