Hausdorff Dimension and Conformal Dynamics I: Strong Convergence of Kleinian Groups

Abstract

This paper investigates the behavior of the Hausdorff dimensions of the limit sets (\Lambda_n) and (\Lambda) of a sequence of Kleinian groups (\Gamma_n \rightarrow \Gamma), where (M = \mathbb{H}^3/\Gamma) is geometrically finite. We show if (\Gamma_n \rightarrow \Gamma) strongly, then: (a) (M_n = \mathbb{H}^3/\Gamma_n) is geometrically finite for all (n \gg 0), (b) (\Lambda_n \rightarrow \Lambda) in the Hausdorff topology, and (c) (H. dim(\Lambda_n) \rightarrow H. dim(\Lambda)), if (H. dim(\Lambda) \geq 1). On the other hand, we give examples showing the dimension can vary discontinuously under strong limits when (H. dim(\Lambda) < 1). Continuity can be recovered by requiring that accidental parabolics converge radially. Similar results hold for higher-dimensional manifolds. Applications are given to quasifuchsian groups and their limits.