Thurston's K=2 Conjecture about Hyperbolic Convex Hulls

I will give an exposition of joint work with David Epstein and Vlad
Markovic.  We regard the extended complex plane as the boundary of the
upper halfspace model of hyperbolic space. We investigate the geometric
relationship between a simply connected region U in the plane and the
relative boundary V of the hyperbolic convex hull of the complement of U.
Here V in upper halfspace lies over U as the dome of a domed stadium lies
over the playing field.  Of particular interest is the case that V is
invariant under a discrete group of isometries. In this case, our work is
about the geometric relationship between the boundary of a hyperbolic
3-manifold and the boundary of its convex core.