This is a question from an old qualifying exam in topology.
Let $S_g$ be the compact orientable surface of genus $g$. Show that $S_g$ has an irregular 3-fold cover when $g>1$.
While the question does not explicitly state it, I am pretty sure we are meant to assume we are looking for a connected cover.
I understand that this amounts to finding a non-normal index 3 subgroup of $$\pi_1(S_g)=\langle a_1,b_1,\dots,a_g,b_g\big|[a_1,b_1]\cdots[a_g,b_g]\rangle.$$
I'm sure there is a group theoretic way to show such a subgroup exists, and I'm certainly interested in hearing those arguments. However, I'm much more interested in a geometric argument or at least understanding of this covering space.
I've toyed around with it and haven't really gotten anywhere. I've tried to construct a non-normal index 3 subgroup by hand, and I've also been looking at the regular covering spaces of $S_g$ (Example 1.41 in Hatcher) and trying to tweak them in some way.