Let

${\Omega}$ be an open simply-connected subset of the Riemann sphere

${S}^{2}$ (regarded as the boundary of hyperbolic 3-space

${\mathbf{H}}^{3}$) and let

$X={S}^{2}\setminus {\Omega}$. (To avoid special cases, we suppose that

${\Omega}\subset \mathbf{C}$ and

${\Omega}\ne \mathbf{C}$.) We can form the hyperbolic convex hull

$\mathcal{C}H\left(X\right)$, and the authors denote the relative boundary of

$\mathcal{C}H\left(X\right)$ in

${\mathbf{H}}^{3}$ by

$\text{Dome}\left({\Omega}\right)$. The study of the geometry of objects such as

$\text{Dome}\left({\Omega}\right)$ was initiated by Thurston who proved for instance that the hyperboloic metric of

${\mathbf{H}}^{3}$ induces a path metric on

$\text{Dome}\left({\Omega}\right)$ which makes it isometric to the hyperbolic disk

${\mathbf{D}}^{2}$. For such an

${\Omega}$, Thurston defined a “nearest point retraction”

$r:{\Omega}\to \text{Dome}\left({\Omega}\right)$ as follows: for each

$z\in {\Omega}$, we expand a small horoball at

$z$ and we call

$r\left(z\right)\in \text{Dome}\left({\Omega}\right)$ the unique first point of contact. Sullivan, and Epstein-Marden analysed that construction and they proved that there exists

$K$ such that for any simply connected

${\Omega}\ne \mathbf{C}$, there is a

$K$-quasiconformal homeomorphism

${\Psi}:\text{Dome}\left({\Omega}\right)\to {\Omega}$ which extends continuously to the identity map on the common boundary

$\partial {\Omega}$. Thurston suggested that the best constant

$K$ is 2, and this suggestion has been called later on “Thurston’s

$K=2$ conjecture”. In this paper, the authors give a counterexample to that conjecture in its equivariant form, that is, in the case where the homeomorphism respects a group of Möbius transformations which preserve

${\Omega}$. Another result that the authors prove in this paper is that the nearest point retraction

$r$ is 2-Lipschitz in the respective hyperbolic metrics, and that the constant 2 here is best possible. The authors also study pleating maps of hyperbolic 2-plane. They obtain explicit universal constants

$0<{c}_{1}<{c}_{2}$ such that no pleating map which bends more than

${c}_{1}$ in some interval of unit length is an embedding, and such that any pleating map which bends less than

${c}_{2}$ in each interval of unit length is embedded. They show that every

$K$-quasiconformal homeomrophism of the unit disk

${\mathbf{D}}^{2}$ is a

$(K,a(K\left)\right)$-quasi-isometry, where

$a\left(K\right)$ is an explicitely computed function, where the multiplicative constant is best possible and where the additive constant

$a\left(K\right)$ is best possible for some values of

$K$.