THE MONODROMY GROUPS OF SCHWARZIAN

EQUATIONS ON COMPACT RIEMANN SURFACES

Daniel Gallo Michael Kapovich Albert Marden

Abstract. Let R be an oriented compact surface without boundary of genus exceeding one, and let θ : π_{1}(R; O) → Γ ⊂ PSL(2, C) be a homomorphism of its fundamental group onto a nonelementary group Γ of Möbius transformations. We present a complete, self-contained proof of the following facts:

(i) θ is induced by a complex projective structure for some complex structure on R if and only if θ lifts to a homomorphism θ^{∗} : π_{1}(R; O) → SL(2, C).

(ii) θ is induced by a branched complex projective structure with a single branch point of order two for some complex structure on R if and only if θ does not lift to a homomorphism into SL(2, C).

(iii) There is a subgroup N of index two in π_{1}(R; O) corresponding to a two-sheeted unbranched cover R̃ of R such that, for some complex structure on R̃, the restriction θ|_{N} is induced by a complex projective structure on R̃ and lifts to a homomorphism θ^{∗} : N → SL(2, C).

Typeset by AMS-TEX