Projective structures, grafting, and measured laminations

TL;DR

This paper demonstrates that grafting a fixed hyperbolic surface creates a homeomorphism between measured laminations and Teichmuller space, and explores the relationship between different coordinate systems for complex projective structures.

Contribution

It establishes a new homeomorphism linking measured laminations and Teichmuller space via grafting, and analyzes the geometric properties of grafting rays in Teichmuller space.

Findings

Grafting defines a homeomorphism between measured laminations and Teichmuller space.

Provides estimates for extremal and hyperbolic length functions along grafting rays.

Clarifies the relationship between complex-analytic and geometric coordinates for projective structures.

Abstract

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.