Introduction

A hyperbolic structure on a surface is related to any other by a left earthquake (to be defined presently). I proved this theorem in the case of closed surfaces several years ago, although I did not publish it at that time.

Stephen Kerckhoff made use of the earthquake theorem in his proof of the celebrated Nielsen Realization Conjecture (Kerckhoff, 1983), and he presented a proof in the appendix to his article.

The original proof in some ways is quite nice, but it has shortcomings. It is not elementary, in that it makes use of the understanding and classification of measured laminations on a surface as well as the classification of hyperbolic structures on the surface. It also uses some basic but non-elementary topology of ℝn. Given this background, the proof is fairly simple, but developed from the ground-up it is complicated and indirect.

In this chapter, I will give a more elementary and more constructive proof of the earthquake theorem. The new proof is inspired by the construction and analysis of the convex hull of a set in space. The newproof also has the advantage that it works in a very general context where the old proof would run into probably unsurmountable difficulties involving infinite-dimensional Teichmüller spaces.

The first part of this chapter will deal with the hyperbolic plane, rather than with a general hyperbolic surface. This will save a lot of fussing over extra definitions and cases.