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

 Title: Hausdorff Dimension and Conformal Dynamics I: Strong Convergence of Kleinian Groups Author: McMullen, Curtis T. Citation: McMullen, Curtis T. 1999. Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups. Journal of Differential Geometry 51(3): 471–515. Full Text & Related Files: hausdorff_strong_convergence.pdf (391.9Kb; PDF) 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. Published Version: http://www.intlpress.com/journals/JDG/archive/vol.51/issue3/3_3.pdf Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:9871959

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