# The monodromy groups of Schwarzian equations in closed Riemann surfaces

### Abstract

The $\theta\colon \pi_1(R)\to \mathrm{PSL}(2,\mathbb{C})$ 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 $\theta$ to be the monodromy representation associated with a complex projective structure on $R$, either unbranched or with a single branch point of order $2$, is that $\theta(\pi_1(R))$ be nonelementary. A branch point is required if and only if the representation) $\theta$ does not lift to $\mathrm{SL}(2,\mathbb{C})$.

Daniel Gallo

Michael Kapovich

Albert Marden