Teichm\"uller theory of the universal hyperbolic lamination

TL;DR

This paper develops a complex analytic model for the Teichmüller space of the universal hyperbolic lamination, establishing a biholomorphic correspondence with a space of continuous functions and exploring its geometric properties.

Contribution

It constructs a new Ahlfors-Bers model for Sullivan's Teichmüller space and identifies natural Kähler coordinates, extending the Weil-Petersson metric to this setting.

Findings

Sullivan's Teichmüller space is Kähler isometric biholomorphic to a function space.

Natural Kähler coordinates for Sullivan's Teichmüller space are identified.

The Nag-Verjovsky embedding is shown to be transversal to Sullivan's space.

Abstract

We construct an Ahlfors-Bers complex analytic model for the Teichm\"uller space of the universal hyperbolic lamination (also known as Sullivan's Teichm\"uller space) and the renormalized Weil-Petersson metric on it as an extension of the usual one. In this setting, we prove that Sullivan's Teichm\"uller space is K\"ahler 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 Teichm\"uller space of this surface; i.e. We find natural K\"ahler coordinates for the Sullivan's Teichm\"uller space. This is the main result. As a corollary, we show the expected fact that the Nag-Verjovsky embedding is transversal to the Sullivan's Teichm\"uller space contained in the universal one.