Is the fundamental group of a compact Riemann surface *after* removing a finite number of points still a Fuchsian group?

Question

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 $\Gamma$?

Answer 1

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 $1$.

Answer 2

To ask whether $\pi_1(S - \mathcal{B})$ is Fuchsian, you first have to think of it as a subgroup of $\operatorname{Aut} \mathbb{H}$, the automorphism group of $\mathbb{H}$ as a complex manifold.

I assume you want to do this by noting that the universal cover of $S - \mathcal{B}$ is isomorphic to $\mathbb{H}$, so the action of the fundamental group on the universal cover becomes an action on $\mathbb{H}$ by automorphisms.

In this case, $\pi_1(S - \mathcal{B})$ is indeed Fuchsian. To see why, recall that the fundamental group of a space always acts evenly (properly discontinuously) on the universal cover. In our case, that means $\pi_1(S - \mathcal{B})$ acts evenly on $\mathbb{H}$. It follows that $\pi_1(S - \mathcal{B})$ is a discrete subgroup of $\operatorname{Aut} \mathbb{H}$, as Borthwick argues at the start of Section 2 of their Introduction to Spectral Theory on Hyperbolic Surfaces.

Answer 3

The surface $S-B$ has a complete hyperbolic metric of finite area. One can see by a pretty direct construction which generalizes the construction of a hyperbolic metric on $S$ itself. Alternatively, since your question is about a given Riemann Surface structure, one can use the uniformization theorem; since removing the points of $B$ creates "removable singularities", the hyperbolic structure in the deleted neighborhood of such a point must be a cusp, which is complete and of finite area.

It follows (from the construction of the "developing map") that the universal cover $\widetilde{S-B}$ is isometric to the hyperbolic plane $\mathbb{H}^2$, and that once an isometry between them is chosen the deck transformation action of $\pi_1(S-B)$ on $\widetilde{S-B} \approx \mathbb{H}^2$ becomes a Fuchsian group.

A good reference for the non-complex-analytic part of this (complete hyperbolic structures; developing maps; etc.) is Thurston's book and the lecture notes that preceded the book.