We prove that if S is a closed compact surface of genus g ≥ 2, and if ρ : π 1 (S) → PSL(2, C) is a quasi-Fuchsian representation, then the deformation space M k,ρ of branched projective structures on S with total branching order k and holonomy ρ is connected, as soon as 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 M k,ρ for non-elementary representations ρ. It is shown to be a smooth complex manifold modeled on Hurwitz spaces.

We prove in two different ways that the monodromy map from the space of irreducible sl 2 -differential-systems on genus two Riemann surfaces, towards the character variety of SL 2 -representations of the fundamental group, is a local diffeomorphism. We also show that this is no longer true in the higher genus case. Our work is motivated by a question raised byÉtienne Ghys about Margulis' problem: the existence of curves of negative Euler characteristic in compact quotients of SL 2 (C). Nous montrons de deux manières différentes que l'application monodromie, depuis l'espace des sl 2 -systèmes différentiels irréductibles sur les surfaces de Riemann de genre deux, vers la variété de caractères des SL 2 -représentations du groupe fondamental, est un difféomorphisme local. Nous montrons aussi que ce n'est plus le cas en genre supérieur. Notre travail est motivé par une question d'Étienne Ghysà propos d'un problème de Margulis : l'existence de courbes de caractéristique d'Euler négative dans les quotients compacts de SL 2 (C).

We prove the connectedness and compute the diameter of the oriented graph of multi-graftings associated to exotic CP 1 -structures on a compact surface S with a given holonomy representation of Fuchsian type.

In this article we investigate the natural domain of definition of a holonomy map associated to a singular holomorphic foliation of P 2 . We prove that germs of holonomy between algebraic curves can have large sets of singularities for the analytic continuation. In the Riccati context we provide examples with natural boundary and maximal sets of singularities. In the generic case we provide examples having at least a Cantor set of singularities and even a nonempty open set of singularities. The examples provided are based on the presence of sufficiently rich contracting dynamics in the holonomy pseudogroup of the foliation.

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