# 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.

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.