For a given quasi-Fuchsian representation ρ : π 1 (S) → PSL 2 C of the fundamental group of a surface S of genus g ≥ 2, we prove that a generic branched complex projective structure on S with holonomy ρ and two branch points is obtained by bubbling some unbranched structure on S with the same holonomy.

We show that the simultaneous (de)grafting of a complex projective structure with quasi-Fuchsian holonomy along a multicurve can be performed by a simple sequence of one bubbling and one debubbling. As a consequence we obtain that any complex projective structure with quasi-Fuchsian holonomy ρ : π 1 (S) → PSL 2 C can be joined to the corresponding uniformizing hyperbolic structure σρ by a simple sequence of one bubbling and one debubbling, with a stopover in the space of branched complex projective structures.

We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More generally we show that Galois Belyĭ curves are precisely those Riemann surfaces for which the Fuchsian uniformization is the unique complex projective structure invariant under the full group of biholomorphisms.

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We study Artin kernels, i.e. kernels of discrete characters of right-angled Artin groups, and we show that they decompose as graphs of groups in a way that can be explicitly computed from the underlying graph. When the underlying graph is chordal, we show that every such subgroup either surjects to an infinitely generated free group or is a generalized Baumslag–Solitar group of variable rank. In particular, for block graphs (e.g. trees), we obtain an explicit rank formula and discuss some features of the space of fibrations of the associated right-angled Artin group.