Complex projective structures with Schottky holonomy

Abstract

Let S be a closed orientable surface of genus at least two. Let Γ be a Schottky group whose rank is equal to the genus of S, and Ω be the domain of discontinuity of Γ. Pick an arbitrary epimorphism ρ : π_1(S) → Γ. Then Ω/Γ is a surface homeomorphic to S carrying a (complex) projective structure with holonomy ρ. We show that every projective structure with holonomy ρ is obtained by (2π-)grafting Ω/Γ along a multiloop on S.

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