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I was wondering what the fundamental group of an infinitely sheeted covering space of say some surface might be?

I'm thinking it should be an infinite cyclic group, but this is more intuitively, i cannot seem to construct an argument for this. Thanks!

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    Do you have anything more specific in mind? The fundamental group of a covering space of $X$ is a subgroup of $\pi_1(X)$, whose index is the number of sheets, so if I take your question literally, the groups you'll get are the infinite-index subgroups of surface groups.2012-12-11
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    Say of a connected sum of two tori?2012-12-11
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    Please add that information *to the question itself*, seria. It is certainly not true that every infinitely sheeted covering of a torus has cyclic group of covering transformations...2012-12-12
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    I'm confused because there is more than one infinitely sheeted covering space. Different coverings have different fundamental groups, so do you mean a particular covering space?2012-12-12

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