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.