Projective structures, grafting, and measured laminations

Understand

We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmuller 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 ($\CP^1$) structures on a surface.
• We also study the rays in Teichmuller space associated to the grafting coordinates, obtaining estimates for extremal and hyperbolic length functions and their derivatives along these grafting rays.