University of Minnesota
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.