Infinitely sheeted covering spaces! 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!
 A: Just for fun, here is an example of an infinite sheeted cover of a genus two surface. The covering map maps each copy of the cylinder with a handle to the rolled-up cylinder where the two ends are identified. You could further unwrap it to a plane with handles attached in a grid pattern to get a covering space with deck transformation group $\mathbb Z\times\mathbb Z$.

A: For the case of a connected sum of two tori, I believe this is equivalent to showing that every non trivial subgroup of the free product $\mathbb{Z}^2\star\mathbb{Z}^2$ is infinite cyclic. Here I am using the fact that the fundamental group of a connected sum of two tori( manifolds) is the free product of their corresponding fundamental groups.
A: An infinitely-sheeted (connected) cover of a surface is necessarily a noncompact surface. Every noncompact surface is homotopy-equivalent to a graph and, hence, has countable free fundamental group. Conversely, every countable free group can be realized as the fundamental group of an infinitely sheeted cover of some compact surface. 
