Mathematics Colloquia and Seminars

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Let S be a closed orientable surface of genus at least two, and let C be an arbitary complex projective structure on S (i.e. a Riemann Surface with transition maps in PSL(2,C)). We show that there is a decomposition of S into pairs of pants and cylinders such that the restriction of C to each component has an injective developing map and a discrete faithful holonomy representation. This decomposition implies that every complex projective structure can be obtained by the construction of Gallo, Kapovich, and Marden. A grafting is an operation on a complex projective structure to obtain a new structure, preserving the holonomy representation. Along the way to prove the decomposition theorem, we show that there is an admissible loop on (S, C), along which a grafting can be done.