A Fr\'echet topology on measured laminations and Earthquakes in the hyperbolic plane

TL;DR

This paper establishes a homeomorphism between bounded measured laminations and the universal Teichmüller space using a Fréchet topology, showing earthquakes with discrete measures are dense and providing infinitesimal versions of these results.

Contribution

It introduces a new Fréchet topology on measured laminations that makes the earthquake correspondence a homeomorphism, advancing the understanding of Teichmüller theory.

Findings

Homeomorphism between $ML_b(\

e9

and $T(\

Abstract

We prove that the bijective correspondence between the space of bounded measured laminations $ML_b(\mathbb{H})$ and the universal Teichm\"uller space $T(\mathbb{H})$ given by $\lambda\mapsto E^{\lambda}|_{S^1}$ is a homeomorphism for the Fr\'echet topology on $ML_b(\mathbb{H})$ and the Teichm\"uller topology on $T(\mathbb{H})$, where $E^{\lambda}$ is an earthquake with earthquake measure $\lambda$. A corollary is that earthquakes with discrete earthquake measures are dense in $T(\mathbb{H})$. We also establish infinitesimal versions of the above results.