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The monodromy groups of Schwarzian equations on closed Riemann surfaces

Daniel Gallo, Michael Kapovich, Albert Marden

math.CVarXiv:math/9511213

Abstract

Let θ:π1(R) (2,) 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(π1(R)) be nonelementary. A branch point is required if and only if the representation θdoes not lift to (2,).

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