Elementary subgroups of surface groups

Chloe Perin in her thesis answered the same question for all torsion-free hyperbolic groups. She classified all the subgroups $H$ of a torsion-free hyperbolic group $G,$ that are elementary submodels of $G.$ If $G$ is a f.g. free group, H must be a non-abelian free factor. If $G$ is a surface group the classification of elementary submodels $H$ of $G$ is slightly more technical and can be found in Chloe's thesis.

Perin's theorem 1.2 gives a necessary condition that is also sufficient in the case of a (closed) surface group (for general torsion-free hyperbolic group a slight modification is needed). A subgroup H is elementary embedded in the fundamental group of a closed surface group S (of genus at least 2), if and only if H is a non-abelian free factor in the fundamental group of a (proper) subsurafce M of S, and there exists a retraction from S onto M. The proof of the necessary part appears in Perin's thesis (theorem 1.2), and the sufficient part follows using my own argument that the elementary core of a torsion-free hyperbolic group is elementary embedded in the hyperbolic group.