Peripheral birationality for 3-dimensional convex co-compact $PSL_2\mathbb{C}$ varieties

It is a consequence of a well-known result of Ahlfors and Bers that the $PSL_2\mathbb{C}$ character associated to a convex co-compact hyperbolic 3-manifold is determined by its peripheral data. In this talk we will show how this map extends to a birational isomorphism of the corresponding $PSL_2\mathbb{C}$ character varieties, so in particular it is generically a 1-to-1 map. Analogous results were proven by Dunfield in the single cusp case, and by Klaff and Tillmann for finite volume hyperbolic 3-manifolds. This is joint work with Ian Agol.