An explicit counterexample to the equivariant $K=2$ conjecture

Abstract:

We construct an explicit example of a geometrically finite Kleinian group $G$ with invariant component $\Omega$ in the Riemann sphere $\textbf {\hat {C}}$ such that any quasiconformal map from $\Omega$ to the boundary of the convex hull of $\textbf {\hat {C}} - \Omega$ in $\textbf {H^3}$ which extends to the identity map on their common boundary in $\textbf {\hat {C}}$, and which is equivariant under the group of Möbius transformations preserving $\Omega$, must have maximal dilatation $K > 2.002$.