The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces

Let θ : π1(R) → PSL(2, ℂ) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. THEOREM. Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that θ(π1(R)) be nonelementary. A branch point is required if and only if the representation θ does not lift to SL(2, ℂ).