Given a plane region \Omega (whose complement contains at least 2 pts),
one can construct the hyperbolic convex hull of its complement. This
will lie in the upper halfspace model of hyperbolic space. It has
one relative boundary component, denoted Dome (\Omega), which lies
over \Omega as the dome of the Metrodome lies over the floor.

My talk will be an exposition of recent joint work with Vlad Markovic.
First, we will discuss a property of plane regions called "uniform
perfectness". Then we will briefly review the construction of hyprbolic
convex hulls. Finally will use the notion of uniform perfectness
to simplify and complete the study of the relation of the hyperbolic 
geometry of Dome(\Omega) to that of \Omega.