Continuity of convex hull boundaries

Abstract

In this paper we consider families of finitely generated Kleinian 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 Gµ 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. © 1995 by Pacific Journal of Mathematics.