Holonomies of hyperbolic surfaces with one branch point

Abstract

Let $S$ be a closed oriented surface of genus $g\geq 2$. We prove that any representation of $π_1(S)$ into $\mathrm{PSL}(2,\mathbb{R})$ with Euler class $3-2g$ is the holonomy of a hyperbolic metric on $S$ with one simple branch point.