Patterson-Sullivan measures and quasi-conformal deformations

Abstract

In this paper we relate the ergodic action of a Kleinian group on the space of line elements to the conformal action of the group on the sphere at infinity. In particular, we show that for a pair of geometrically isomorphic convex co-compact Kleinian groups, the ratio of the length of the Patterson- Sullivan measure on line element space to the length of its push-forward is bounded below by the ratio of the Hausdorff dimensions of the limit sets. Our primary techniques come from ergodic theory and Patterson Sullivan theory.