Fixed points of compositions of earthquakes

Abstract

Let $S$ be a closed surface of genus at least $2$, and let $\lambda$ and $\mu$ be two laminations that fill $S$. Let $E_r^{\lambda }$ and $E_r^{\mu }$ be the right earthquakes on $\lambda$ and $\mu$, respectively. We show that the composition $E_r^{\lambda }\circ E_r^{\mu }$ has a fixed point in the Teichmüller space of $S$. Another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic anti-de Sitter (AdS) manifold. The proof uses some estimates from the geometry of those AdS manifolds.