For any finite subset $B\subset \mathbf{P}^1$, the fundamental group of the Riemann surface $\mathbf{P}^1-B$ is finitely generated.
Is this true if we replace $\mathbf P^1 $ by a higher genus compact connected Riemann surface?
More precisely, let $B\subset X$ be a finite subset of a compact connected Riemann surface $X$ of genus $g>0$. Is the fundamental group of $X-B$ finitely generated?