Pair-correlation density formula for hyperbolic lattice angles

Let $\Gamma$ be a discrete subgroup of $\mathrm{PSL}_2(\mathbb R)$ with fundamental domain of finite area $V_\Gamma$. For a fixed $\omega\in\mathbb H$, write $\ell(M)=d(\omega,M\omega)$ for $M\in\Gamma$. The pair-correlation measure $R_2(\xi)$ is defined on $[0,\infty)$ as in the preceding setup.

Pair-correlation density conjecture. The pair-correlation measure $R_2(\xi)$ exists on $[0,\infty)$ and is given by a $C^1$ function expressed as a series of three-dimensional volumes. Its density satisfies

$$g_2\left(\frac{\xi}{V_\Gamma}\right)=\frac{V_\Gamma}{\pi\xi^2}\sum_{M\in\Gamma}f_\xi\bigl(\ell(M)\bigr),$$

where, for $\ell\geq 0$ and $\xi>0$,

$$f_\xi(\ell)= \begin{cases} \displaystyle \ln\left(\frac{\cosh\ell+\sinh\ell}{\cosh\ell+\sqrt{\sinh^2\ell-\xi^2}}\right),&\text{if }\xi\leq 2\sinh\left(\frac{\ell}{2}\right),\\ \displaystyle \ln\left(\frac{(\cosh\ell+\sinh\ell)(1+\xi^2)}{(\cosh\ell+\sqrt{\sinh^2\ell-\xi^2})^2}\right),&\text{if }2\sinh\left(\frac{\ell}{2}\right)\leq\xi\leq\sinh\ell,\\ \ln(\cosh\ell+\sinh\ell)=\ell,&\text{if }\sinh\ell\leq\xi. \end{cases}$$

This gives an explicit formula for pair correlations of directions of hyperbolic lattice points for arbitrary discrete finite-covolume subgroups, extending the established modular-group cases and expressing the density through an automorphic-kernel-type series.