Meromorphic projective structures, grafting and the monodromy map

A meromorphic projective structure on a punctured Riemann surface X∖P is determined, after fixing standard X, by quadratic differential with poles of order three or more at each puncture in P. In this article we prove the analogue Thurston's grafting theorem for such structures, that involves crowned hyperbolic surfaces. This also provides description structures C have polynomial Schwarzian derivatives. As an application our main result, result Hejhal, namely, show monodromy map to decorated character variety (in sense Fock-Goncharov) local homeomorphism.