The Monodromy Groups of Schwarzian Equations on Compact Riemann Surfaces

Let R be an oriented compact surface without boundary of genus exceeding one, and let θ : π1(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∗ : π1(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 π1(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). This research was done in part at the Mathematics Institute of the University of Warwick and the Forschungsinstitut für Mathematik at ETH, Zürich (Marden), and at the Mathematical Sciences Research Institute in Berkeley (Marden and Kapovich). The authors acknowledge support from the NSF grants DMS-9306140 (Kapovich) and DMS-9022140 (Kapovich and Marden). See also page 5.