Continuity of convex hull boundaries

Linda Jo Goldway Keen and Caroline Series

Vol. 168 (1995), No. 1, 183–206

DOI: 10.2140/pjm.1995.168.183

Abstract

In this paper we consider families of finitely generated Kleinian groups {G_μ} that depend holomorphically on a parameter μ which varies in an arbitrary connected domain in ℂ. The groups G_μ are quasiconformally conjugate. We denote the boundary of the convex hull of the limit set of G_μ by ∂𝒞(G_μ). The quotient ∂𝒞(G_μ)∕G_μ is a union of pleated surfaces each carrying a hyperbolic structure. We fix our attention on one component S_μ 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 S_μ are continuous functions of μ.

Mathematical Subject Classification 2000

Primary: 30F40

Secondary: 57M50

Milestones

Received: 13 August 1992

Revised: 12 April 1993

Published: 1 March 1995