Let $\Sigma$ be a closed orientable surface of genus $g \geq 2$. Suppose that $\Sigma = \partial H$ where $H$ is a handlebody. We then have the subgroup $N$ which is the kernel of the inclusion $\pi_1(\Sigma) \to \pi_1(H)$. I know that this map is surjective on $\pi_1$ and since $\pi_1(H)$ is a free group of rank $g$, $N$ must be a countably generated free group.
What is the topology of the surface $\Sigma'$ covering $\Sigma$ corresponding to $N$? I imagine that since the group must be free, there can not be any genus. A reference for the result would be wonderful.
I know that orientable open surfaces are classified by their genus (either finite or infinite) and their ideal boundary. More generally I would love to know which of these types of surfaces can occur as (regular and/or irregular?) coverings of $\Sigma$.