Pair correlation of hyperbolic lattice angles

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.

28 pages. Final version