# Are fundamental groups of Riemann surfaces always finitely generated

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?

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I don't think this is true. But it is true that they're countable. – mixedmath Mar 5 '12 at 17:55

The complement of a non-empty finite set in a closed (real!) manifold of dimension $2$ has the homotopy type of a finite graph, so the answer is yes.