Volume 19, issue 6 (2015)

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

Shinpei Baba

Geometry & Topology 19 (2015) 3233–3287
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 $\mathsc{G}\mathsc{ℒ}$ of geodesic laminations on $S$ via a natural projection of projective structures on $S$ into $\mathsc{G}\mathsc{ℒ}$ in Thurston coordinates. In a sequel paper, using this local solution, we prove the conjecture for generic holonomy.

Keywords
surface, complex projective structure, holonomy, grafting
Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 30F40, 20H10