Quasiconformal Lipschitz maps, Sullivan's convex hull theorem and Brennan's conjecture

We show that proving the conjectured sharp constant in a theorem of Dennis Sullivan concerning convex sets in hyperbolic 3-space would imply the Brennan conjecture. We also prove that any conformal map f: D --> Omega can be factored as a K-quasiconformal self-map of the disk (with K independent of Omega) and a map g: D --> Omega with derivative bounded away from zero. In particular, there is always a Lipschitz homeomorphism from any simply connected Omega (with its internal path metric) to the unit disk.