Dimension of escaping geodesics

Suppose $M=\mathbb {B}/G$ is a hyperbolic manifold. Consider the set of escaping geodesic rays $\gamma (t)$ originating at a fixed point $p$ of the manifold $M$, i.e. $\operatorname {dist}(\gamma (t),p)\to \infty$. We investigate those escaping geodesics which escape at the fastest possible rate, and find the Hausdorff dimension of the corresponding terminal points on the boundary of $\mathbb {B}$.

In dimension $2$, for a geometrically infinite Fuchsian group, if the injectivity radius of $M=\mathbb {B}/G$ is bounded above and away from zero, then these points have full dimension. In dimension $3$, when $G$ is a geometrically infinite and topologically tame Kleinian group, if the injectivity radius of $M=\mathbb {B}/G$ is bounded away from zero, the dimension of these points is $2$, which is again maximal.