#### Volume 19, issue 6 (2015)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
$2\pi$–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
##### Publication
Received: 3 February 2014
Revised: 22 November 2014
Accepted: 26 January 2015
Published: 6 January 2016
Proposed: Benson Farb
Seconded: Jean-Pierre Otal, Danny Calegari
##### Authors
 Shinpei Baba Ruprecht-Karls-Universität Heidelberg Mathematisches Institut Im Neuenheimer Feld 368 Heidelberg, D-69120 Germany