Daniel Gallo and Michael Kapovich and Albert Marden
\def\PSL{\hbox{PSL}} \def\SL{\hbox{SL}} \def\C{{\bold C}} \def\m/{M\"obius} Let $R$ be an oriented compact surface without boundary of genus exceeding one, and let $ \theta:\pi_1(R;O)\to \Gamma\subset\PSL(2,\C) $ be a homomorphism of its fundamental group onto a nonelementary group $\Gamma$ of \m/ transformations. We present a complete, self-contained proof of the following facts:
\smallskip \hangindent1em\hangafter1 \noindent (i) $\theta$ 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\SL(2,\C). $
\smallskip \hangindent1em\hangafter1 \noindent (ii) $\theta$ 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 $\SL(2,\C)$.
\smallskip \hangindent1em\hangafter1 \noindent (iii) 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 $\theta^*:N\to\SL(2,\C)$.