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. 1 Basics and Statement of Results Let Isom+(H) n ≥ 2 be the space of orientation-preserving isometries of Hn. As is well-known, this space of isometries can be given the topology induced by uniform convergence on compact sets. A Kleinian group Γ is a discrete subgroup of Isom+(H). As such, Γ acts discontinuously on Hn, and because we make a standing assumption that the action is torsion-free, the quotient manifoldN = Hn/Γ is a complete Riemannian manifold of constant curvature −1. A Kleinian group Γ also acts as a discrete subgroup of conformal automorphisms of the sphere at infinity Sn−1 ∞ ; this action partitions S n−1 ∞ into two disjoint sets. The regular set ΩΓ is the largest open set in Sn−1 ∞ on which Γ acts properly discontinuously, and the limit set LΓ is its complement. In the case that LΓ contains more than 2 points, the limit set is characterized as being the smallest closed Γ-invariant subset of Sn−1 ∞ . Define the convex hull CH(LΓ) of the limit set LΓ to be the smallest convex subset of Hn so that all geodesics with both limit points in LΓ are contained in CH(LΓ). We can take the quotient of CH(LΓ) by Γ (denoted by C(Γ)); this is