2 pi-grafting and complex projective structures, I

摘要

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 GL of geodesic laminations on S via a natural projection of projective structures on S into gl in Thurston coordinates. In a sequel paper, using this local solution, we prove the conjecture for generic holonomy.