The logarithmic spiral: a counterexample to the K = 2 conjecture

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 analytic, 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.