Projective structures, grafting and measured laminations

Abstract

We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmüller space, complementing a result of Scannell–Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complex-analytic and geometric coordinate systems for the space of complex projective (${ℂ\mathbb{P}}^1$) structures on a surface.

We also study the rays in Teichmüller space associated to the grafting coordinates, obtaining estimates for extremal and hyperbolic length functions and their derivatives along these grafting rays.