Infinitely sheeted covering spaces!

Question

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!

Answer 1

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.