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The monodromy groups of Schwarzian equations on closed Riemann surfaces

  • Author(s): Gallo, D;
  • Kapovich, M;
  • Marden, A
  • et al.

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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).

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