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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.

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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.

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  • $\begingroup$ I think another useful reference is Proposition 9.2 of arxiv.org/abs/1003.4095 . $\endgroup$
    – HJRW
    Commented Mar 31, 2017 at 8:54
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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.

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  • $\begingroup$ @YCor I believe the OP attempted to provide the same link. $\endgroup$
    – Igor Rivin
    Commented Mar 31, 2017 at 1:40
  • $\begingroup$ @YCor if you look at the original post, you will see a reference to a Theorem 1.4 and a link. If you click on the link, you will see that, while it is broken, it is clearly meant to point at the exact thing you have posted a correct link to. $\endgroup$
    – Igor Rivin
    Commented Mar 31, 2017 at 2:21
  • $\begingroup$ @YCor I have mentioned this theorem in the last line of the first paragraph. This link was provided exactly before the last word ("this"). Unfortunately the link was not working. Thank you very much for providing the link and clarifying he matter. $\endgroup$
    – Cusp
    Commented Mar 31, 2017 at 2:30
  • $\begingroup$ @YCor My point precisely. $\endgroup$
    – Igor Rivin
    Commented Mar 31, 2017 at 2:40
  • $\begingroup$ Finally the new answer (perhaps it would have been better edit this one) clarifies the matter. I erased my contribution to a now obsolete conversation. $\endgroup$
    – YCor
    Commented Mar 31, 2017 at 14:35

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