# 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