Is the fundamental group of a compact Riemann surface *after* removing a finite number of points still a Fuchsian group? Let $S$ be a compact R.S. admitting a Fuchsian model $\mathbb{H} / \Gamma$. We know that $\pi_1(S) \cong \Gamma$. Let $\mathcal{B} \subseteq S$ be a finite set of points, is $\pi_1(S - \mathcal{B})$ still a Fuchsian group? How is it related to $\Gamma$?

Yes; a hyperbolic surface minus a finite set of points is still a hyperbolic surface. If $S$ has genus $g$ then the fundamental group of $S$ minus a point, as an abstract group, is free on $2g$ generators (the same $2g$ generators generating $\pi_1(S)$, but no longer constrained by their usual relation), and every point removed after that increases the number of free generators by $1$. 
