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Math Help - Universal cover of n times punctured plane

  1. #1
    MHF Contributor Bruno J.'s Avatar
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    Universal cover of n times punctured plane

    I'm looking for some insight into the relationship between the complex plane punctured 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 \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 \lambda=k^2 being the corresponding uniformizing function (whose automorphism group \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 n-times punctured plane, and what is the corresponding uniformizing function? I suppose that the universal cover is the upper-half plane for n\geq 2, with a modular function as the uniformizing function. However, this would imply that \mbox{PSL}(2, \mathbb{Z}) contains a copy of the free group on 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!
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  2. #2
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    I don't know enough algebraic topology to answer this question, but I do know that \mathbb{F}_2, the free group on two generators, contains a copy of \mathbb{F}_n as a subgroup, and therefore so does \text{PSL}(2,\mathbb{Z}). Here, n can be any positive integer or even infinity. If a, b are generators of \mathbb{F}_2 then (if I remember correctly) you can take a^kb^ka^k\ (1\leqslant k\leqslant n) as generators for a copy of \mathbb{F}_n.
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by Opalg View Post
    I don't know enough algebraic topology to answer this question, but I do know that \mathbb{F}_2, the free group on two generators, contains a copy of \mathbb{F}_n as a subgroup, and therefore so does \text{PSL}(2,\mathbb{Z}). Here, n can be any positive integer or even infinity. If a, b are generators of \mathbb{F}_2 then (if I remember correctly) you can take a^kb^ka^k\ (1\leqslant k\leqslant n) as generators for a copy of \mathbb{F}_n.
    That's awesome! So I guess the possibility of the uniformizing function being a modular function is not ruled out.
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    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
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