Teichmuller theory of the universal hyperbolic lamination

We construct an Ahlfors-Bers complex analytic model for the Teichmuller space of the universal hyperbolic lamination (also known as Sullivans Teichmuller space) and the renormalized Weil-Petersson metric on it as an extension of the usual one. In this setting, we prove that Sullivans Teichmuller space is Kahler isometric biholomorphic to the space of continuous functions from the profinite completion of the fundamental group of a compact Riemann surface of genus greater than or equal to two to the Teichmuller space of this surface; i.e. We find natural Kahler coordinates for the Sullivans Teichmuller space. This is the main result. As a corollary, we show the expected fact that the Nag-Verjovsky embedding is transversal to the Sullivans Teichmuller space contained in the universal one.