Institute for Mathematical Sciences

Preprint ims90-16

L. Keen and C. Series
Continuity of Convex Hull Boundaries

Abstract: In this paper we consider families of finitely generatedKleinian groups {Gμ} that depend holomorphically on a parameter μ which varies in an arbitrary connected domain in C. The groups Gμ are quasiconformally conjugate. We denote the boundary of the convex hull of the limit set of G\EC by ∂C{Gμ). The quotient ∂C(Gμ)/Gμ is a union of pleated surfaces each carrying a hyperbolic structure. We fix our attention on one component Sμ and we address the problem of how it varies with μ. We prove that both the hyperbolic structure and the bending measure of the pleating lamination of Sμ are continuous functions of μ.
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