Universal cover of n times punctured plane

I'm looking for some insight into the relationship between the complex plane punctured $\displaystyle n$ times and its universal cover. I understand that the universal cover of the once-punctured plane is the plane itself, with the corresponding uniformizing function being the exponential (whose automorphism group $\displaystyle \cong \mathbb{Z}$ is isomorphic to the fundamental group of the base and to the group of deck transformations of the cover). I understand also that that the universal cover of the twice punctured plane is the unit disc (or upper half-plane), with the elliptic modular function $\displaystyle \lambda=k^2$ being the corresponding uniformizing function (whose automorphism group $\displaystyle \cong \mbox{free group on two generators} \cong \Gamma(2) \triangleleft \mbox{PSL}(2, \mathbb{Z})$) is once again isomorphic to the fundamental group of the base, and to the group of deck transformations of the cover).

In general, what is the universal cover of the $\displaystyle n$-times punctured plane, and what is the corresponding uniformizing function? I suppose that the universal cover is the upper-half plane for $\displaystyle n\geq 2$, with a modular function as the uniformizing function. However, this would imply that $\displaystyle \mbox{PSL}(2, \mathbb{Z})$ contains a copy of the free group on $\displaystyle n$ generators as a subgroup, which I doubt very much! It's impressive enough that it contains a copy of the free group on *two* generators...

Any pointers are greatly appreciated! (Nod)

Re: Universal cover of n times punctured plane

Hi Bruno,

Did you find a good resolution to this question? It's something that I am quite interested in as well. (I've heard the term Schottky space and Schottky group come up in this context)

Ralph