Communications in Analysis and Geometry
Volume 13 (2005)
Patterson-Sullivan measures and quasi-conformal deformations
Pages: 561 – 589
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.