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# 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

#### 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