Let $G=\pi_1 (S)$ for a closed surface $S$, consider a closed geodesic $c$ on $S$ and let $H$ be the subgroup of $G$ induced by $c$. Is it true that $G$ splits over $H$, i.e. $G$ can be written as a free product amalgamated over $H$ or as an HNN extension over $H$?
It is clear to me that the answer is positive when $S$ is a torus and $c$ one of the standard generators of $\pi_1 (S)$ and also when $S$ is a surface of higher genus and $c$ is a separating closed geodesic: in the first case we get an HNN extension and in the second case the free product of the two components of $S \setminus c$ amalgamated over $H$.
In the case $c$ is non separating and $S$ has genus $g\geq 2$, the covering space associated to $H$ is a cylinder (this of course holds as well in the above cases), so $H$ is a codimension 1 subgroup of $G$; but this in general is not enough to conclude that $G$ splits over $H$. Anyway, is it enough in this context?
Just sharing what I found: the answer was yes at least when the curve is 2-sided (which comes for free if the surface is orientable), according to this link. No need to talk about codimension-1 subgroups, it's just the old good Seifert Van Kampen. I found this answer to another question very useful.