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Branched projective structures with Fuchsian holonomy

Gabriel Calsamiglia, Bertrand Deroin and Stefano Francaviglia

Geometry & Topology 18 (2014) 379–446

We prove that if S is a closed compact surface of genus g 2, and if ρ: π1(S) PSL(2, ) is a quasi-Fuchsian representation, then the space k,ρ of branched projective structures on S with total branching order k and holonomy ρ 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 k,ρ for non-elementary representations ρ. It is shown to be a smooth complex manifold modeled on Hurwitz spaces.

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