On the distribution of angles between geodesic rays associated with hyperbolic lattice points

Details On the distribution of angles between geodesic rays associated with hyperbolic lattice points On the distribution of angles between geodesic rays associated with hyperbolic lattice points

2006-08-03

arxiv.org/abs/math/0608078v2

Mathematics

Number Theory

12 pages

Scientific paper

For every two points $z_0,z_1$ in the upper half-plane, consider all elements $\gamma$ in the principal congruence group $\Gamma(N)$, acting on the upper half-plane by fractional linear transformations, such that the hyperbolic distance between $z_1$ and $\gamma z_0$ is at most $R>0$. We study the distribution of angles between the geodesic rays $[z_1,\gamma z_0]$ as $R\to \infty$, proving that the limiting distribution exists independently of $N$ and explicitly computing it. When $z_1=z_0$ this is found to be the uniform distribution on the interval $[-\pi/2,\pi/2]$.

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