Teichmüller theory via random simple closed curves

In this talk, we show that the map σ:Tg→Cg, which sends a compact hyperbolic surface X to a random simple closed geodesics on X, defines a proper embedding of Teichmüller space into the space of geodesic currents. In particular, we show that σ induces a compactification of Tg by projective measured laminations that agrees with Thurston’s compactification almost everywhere but differs from it at infinitely many points. This is joint work with Curt McMullen.