$2\pi$–grafting and complex projective structures, I

Abstract

Let $S$ be a closed oriented surface of genus at least two. Gallo, Kapovich and Marden asked whether $2\pi$–grafting produces all projective structures on $S$ with arbitrarily fixed holonomy (the Grafting conjecture). In this paper, we show that the conjecture holds true “locally” in the space $\mathcal{G}\mathcal{ℒ}$ of geodesic laminations on $S$ via a natural projection of projective structures on $S$ into $\mathcal{G}\mathcal{ℒ}$ in Thurston coordinates. In a sequel paper, using this local solution, we prove the conjecture for generic holonomy.