#### Volume 18, issue 1 (2014)

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Branched projective structures with Fuchsian holonomy

### Gabriel Calsamiglia, Bertrand Deroin and Stefano Francaviglia

Geometry & Topology 18 (2014) 379–446
##### Abstract

We prove that if $S$ is a closed compact surface of genus $g\ge 2$, and if $\rho :{\pi }_{1}\left(S\right)\to PSL\left(2,ℂ\right)$ is a quasi-Fuchsian representation, then the space ${\mathsc{ℳ}}_{k,\rho }$ of branched projective structures on $S$ with total branching order $k$ and holonomy $\rho$ is connected, for $k>0$. Equivalently, two branched projective structures with the same quasi-Fuchsian holonomy and the same number of branch points are related by a movement of branch points. In particular grafting annuli are obtained by moving branch points. In the appendix we give an explicit atlas for ${\mathsc{ℳ}}_{k,\rho }$ for non-elementary representations $\rho$. It is shown to be a smooth complex manifold modeled on Hurwitz spaces.

##### Keywords
projective structures, fuchsian holonomy, moduli spaces
##### Mathematical Subject Classification 2010
Primary: 30F35, 57M20
Secondary: 53A30, 14H15
##### Publication
Received: 16 April 2012
Accepted: 30 May 2013
Published: 29 January 2014
Proposed: Benson Farb
Seconded: Yasha Eliashberg, Jean-Pierre Otal
##### Authors
 Gabriel Calsamiglia Instituto de Matemática Universidade Federal Fluminense Rua Mário Santos Braga s/n 24020-140, Niterói Brazil Bertrand Deroin Département de mathématiques d’Orsay Université Paris 11 91405 Orsay Cedex France Stefano Francaviglia Dipartimento di Matematica Università di Bologna P.zza Porta S. Donato 5 40126 Bologna Italy