2 π-grafting and complex projective structures, I

Abstract

Let $S$ be a closed oriented surface of genus at least two. Gallo, Kapovich, and Marden asked if 2π-graftings produce all projective structures on $S$ with arbitrarily fixed holonomy (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 the Thurston coordinates. In the sequel paper, using this local solution, we prove the conjecture for generic holonomy.

57 pages, 10 figures. To appear in Geometry & Topology

References in corpus (1)

• The oriented graph of multi-graftings in the Fuchsian case

Cited by in corpus (5)

• Monodromy groups of $\mathbb{C}{\mathrm{P}}^1$-structures on punctured surfaces
• Bubbling complex projective structures with quasi-Fuchsian holonomy
• Local deformations of branched projective structures: Schiffer variations and the Teichmüller map
• Holonomy map fibers of $\mathbb{C}{\mathrm{P}}^1$-structures in moduli space
• Tame and relatively elliptic ${\mathbb{CP}}^1$-structures on the thrice-punctured sphere