The Monodromy Groups of Schwarzian Equations on Compact Riemann Surfaces
Daniel Gallo and Michael Kapovich and Albert Marden
Let $R$ be an oriented compact surface without boundary of genus exceeding one, and let $\theta:\pi_1(R;O)\to \Gamma\subset{\rm PSL}(2,{\Bbb C})$ be a homomorphism of its fundamental group onto a nonelementary group $\Gamma$ of Möbius/ transformations. We present a complete, self-contained proof of the following facts:
- $\theta$ is induced by a complex projective structure for some complex structure on $R$ if and only if $\theta$ lifts to a homomorphism $\theta^*:\pi_1(R;O)\to{\rm SL}(2,{\Bbb C}).$
- $\theta$ 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 $\theta$ does not lift to a homomorphism into ${\rm SL}(2,{\Bbb C})$.
- There is a subgroup $N$ of index two in $\pi_1(R;O)$ corresponding to a two-sheeted unbranched cover $\tilde R$ of $R$ such that, for some complex structure on $\tilde R$, the restriction $\theta|_N$ is induced by a complex projective structure on $\tilde R$ and lifts to a homomorphism $\theta^*:N\to{\rm SL}(2,{\Bbb C})$.