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    $\begingroup$ Is there a more direct relation between these two proofs, other than that they end in a similar way? E.g., is there a correspondence or a relation between the conformal structure on the Jordan domain at infinity, and the hyperbolic structure on the pleated surface? $\endgroup$ Commented Mar 22, 2011 at 7:46
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    $\begingroup$ Dennis Sullivan found a simple way to construct a map from the boundary of the convex hull of a simply-connected domain to the Poincaré metric on a simply-connected domain that has a bounded bi-Lipschitz constant. David Epstein and Al Marden subsequently gave a very detailed and explicit construction, with an explicit constant. The natural conjecture (or at least a conjecture that is attributed to me) was that the best quasiconformal constant should be 2, but this was disproved: cf. Epstein, Marden and Markovic, Annals of Math 159 (2004) pp 305--336. $\endgroup$ Commented Mar 24, 2011 at 1:29