  In this paper, we first prove that the cubic, defocusing nonlinear
Schr\"odinger equation on the two dimensional hyperbolic space with radial
initial data in $H^s(\mathbb{H}^2)$ is globally well-posed and scatters when $s
> \frac{3}{4}$. Then we extend the result to nonlineraities of order $p>3$. The
result is proved by extending the high-low method of Bourgain in the hyperbolic
setting and by using a Morawetz type estimate proved by the first author and
Ionescu.
