From Sela's proof of Tarski's conjecture we know that the surface groups (i.e. fundamental group of a closed surface of genus $\geq 2$) and free (non-abelian) groups have the same first order theory. We also know that the inclusion of a free non-abelian group into another as a free factor is an elementary embedding. Now any subgroup $H$ of a surface group $G$ is either a surface group (if the index is finite) or a free group (if the index is infinite). Hence the elementary theory of $H$ is equivalent to that of $G$. On the other hand not every inclusion of a subgroup of the surface group is an elementary embedding. A necessary condition is provided in Theorem 1.4 of [this](https://arxiv.org/abs/0903.0945) paper. 


**Q) Is there a sufficient condition for the inclusion to be an elementary embedding?**


**Q) Can we completely classify the subgroups of surface groups for which the inclusion is an elementary embedding?**


I am a newcomer in this area and I am currently reading Sela's proof of the above theorems therefore any suggestion about papers or references regarding the problems will be extremely helpful. Thanks in advance. 



Elementary subgroups of surface groups
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. 


