Given a nonempty compact connected subset X ⊂ S 2 with complement a simply-connected open subset Ω ⊂ S 2 , let Dome(Ω) be the boundary of the hyperbolic convex hull in H 3 of X. We show that if X is a certain logarithmic spiral, then we obtain a counterexample to the conjecture of Thurston and Sullivan that there is a 2-quasiconformal homeomorphism Ω → Dome(Ω) which extends to the identity map on their common boundary in S 2 . This leads to related counterexamples when the boundary is real

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... tic, or a finite union of intervals (straight intervals, if we take S 2 = C ∪ {∞}). We also show how this counterexample enables us to construct a related counterexample which is a domain of discontinuity of a torsion-free quasifuchsian group with compact quotient. Another result is that the average long range bending of the convex hull boundary associated to a certain logarithmic spiral is approximately .98π/2, which is substantially larger than that of any previously known example.