The Anti-de Sitter proof of Thurston's earthquake theorem

Abstract

Thurston's earthquake theorem asserts that every orientation-preserving homeomorphism of the circle admits an extension to the hyperbolic plane which is a (left or right) earthquake. The purpose of these notes is to provide a proof of Thurston's earthquake theorem, using the bi-invariant geometry of the Lie group $\mathrm{PSL}(2,\mathbb{R})$, which is also called Anti-de Sitter three-space. The involved techniques are elementary, and no background knowledge is assumed apart from some two-dimensional hyperbolic geometry.

28 pages, 4 figures. This article will appear as a chapter in the book: In the tradition of Thurston, III (ed. K. Ohshika and A. Papadopoulos), Springer Verlag, 2023