Pure and Applied Mathematics Quarterly

Volume 9 (2013)

Number 1

Special Issue: In Honor of Dennis Sullivan, Part 1 of 2

Uniformly perfect domains and convex hulls: Improved bounds in a generalization of a theorem of Sullivan

Pages: 49 – 71

DOI: http://dx.doi.org/10.4310/PAMQ.2013.v9.n1.a2

Authors

Martin Bridgeman (Department of Mathematics, Boston College, Chestnut Hill, Massachusetts, U.S.A.)

Richard D. Canary (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)

Abstract

Given a hyperbolic domain $\Omega$, the nearest point retraction is a conformally natural homotopy equivalence from $\Omega$ to the boundary Dome($\Omega$) of the convex core of its complement. Marden and Markovic showed that if $\Omega$ is uniformly perfect, then there exists a conformally natural quasiconformal map from $\Omega$ to Dome($\Omega$) which admits a bounded homotopy to the nearest point retraction. We obtain an explicit upper bound on the quasiconformal dilatation which depends only on the injectivity radius of the domain.

Keywords

uniformly perfect domains, convex hulls, Kleinian groups

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