Elementary subgroups of surface groups 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 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.    
 A: 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.
A: 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. 
