Length distortion and the Hausdorff dimension of limit sets

Let Gamma be a convex co-compact quasi-Fuchsian Kleinian group. We define t he distortion function along geodesic rays lying on the boundary of the con vex hull of the limit set, where each ray is pointing in a randomly chosen direction. The distortion function measures the ratio of the intrinsic to e xtrinsic metrics, and is defined asymptotically as the length of the ray go es to infinity. Our main result is that the distortion function is both alm ost everywhere constant and bounded above by the Hausdorff dimension of the limit set of Gamma. As a consequence, we are able to provide a geometric p roof of the following result of Bowen: If the limit set of Gamma is not a r ound circle, then the Hausdorff dimension of the limit set is strictly grea ter than one. The proofs are developed from results in Patterson-Sullivan t heory and ergodic theory.