The Anti-de Sitter Proof of Thurston’s Earthquake Theorem

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.