Branched projective structures with Fuchsian holonomy

Calsamiglia, Gabriel¹ ; Deroin, Bertrand² ; Francaviglia, Stefano³

¹ Instituto de Matemática, Universidade Federal Fluminense, Rua Mário Santos Braga s/n, 24020-140, Niterói, Brazil
² Département de mathématiques d’Orsay, Université Paris 11, 91405 Orsay Cedex, France
³ Dipartimento di Matematica, Università di Bologna, P.zza Porta S. Donato 5, 40126 Bologna, Italy

Geometry & topology, Tome 18 (2014) no. 1, pp. 379-446.

Résumé

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 ${\mathcal{ℳ}}_{k,\rho }$ 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 ${\mathcal{ℳ}}_{k,\rho }$ for non-elementary representations $\rho$. It is shown to be a smooth complex manifold modeled on Hurwitz spaces.

Détail

DOI : 10.2140/gt.2014.18.379

Classification : 30F35, 57M20, 53A30, 14H15
Keywords: projective structures, fuchsian holonomy, moduli spaces

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