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

Abstract

Given a nonempty compact connected subset $X\subset\mathbb{S}^2$ with complement a simply-connected open subset $\Omega\subset\mathbb{S}^2$, let $\mathrm{Dome}({\Omega})$ be the boundary of the hyperbolic convex hull in $\mathbb{H}^3$ 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 $\Omega\to\mathrm{Dome}({\Omega})$ which extends to the identity map on their common boundary in $\mathbb{S}^2$. This leads to related counterexamples when the boundary is real analytic, or a finite union of intervals (straight intervals, if we take $\mathbb{S}^2=\mathbb{C}\cup\left\{\infty\right\}$). 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 $.98\pi/2$, which is substantially larger than that of any previously known example.

Authors

David B. A. Epstein

Mathematics Institute
University of Warwick
Coventry CV4 7AL
United Kingdom

Vladimir Markovic

Mathematics Institute
University of Warwick
Coventry CV4 7AL
United Kingdom