Constructing a surface cover explicitly given a set of generators

Question

Let $F$ be a closed orientable surface. If I am explicitly given a set of elements $g_1,...,g_k \in \pi_1(F)$ (say as curves on the surface or words in the standard set of generators). How can I construct the covering $\tilde{F} \to F$ corresponding to the subgroup $\left< g_1,...,g_k \right>$?

Answer 1

I'm not sure which covering space you're looking for, but one way to produce a covering space is as follows.

Take the universal cover $\pi: X \to F$. Then $\pi_1(F)$ acts on each fiber $\pi^{-1}(x)$ via the monodromy action, so in particular any subgroup $G$ of $\pi_1(F)$ preserves $\pi^{-1}(x)$. Fixing $E=\pi^{-1}(x_0)$ we get an associated fiber bundle $F'=X \times_G E \to F$ obtained by taking the quotient of $X \times E$ by the diagonal action of $G$.