The monodromy groups of Schwarzian equations on closed Riemann surfaces

Abstract

Let θ:π_1(R) \to \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 θto 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(π_1(R)) be nonelementary. A branch point is required if and only if the representation θdoes not lift to \SL(2,\C).