Realization of subgroups of $\pi_1(M)$ by submanifolds Let $M$ be a smooth oriented connected 3-manifold, let $G$ be the fundamental group $\pi_1(M)$. If $M$ is compact, by Kneser-Milnor Decomposition Theorem $M$ can be written as a connected sum of prime closed oriented 3-manifolds $$M=M_1 \sharp \ldots \sharp M_k,$$
then we have that
$$\pi_1(M)=\pi_1(M_1) * \ldots * \pi_1(M_k),$$ and each factor of the free product can be realized by a submanifold, i.e. there exists a compact embedded 3-submanifold with boundary whose fundamental group is $\pi_1(M_i)$ (namely, $M_i$ with an open disc removed). Is there a "similar" construction for a generic subgroups of $\pi_1(M)$?
More specifically, let us suppose that $M$ is non-compact and let $H$ be a finitely generated subgroup of $G$. Under which assumptions (if any) there exists a (compact with boundary) embedded 3-submanifold $N$ of $M$ with $\pi_1(N)=H$ and such that the embedding induces an injection between the fundamental groups $\pi_1(N) \hookrightarrow \pi_1(M)$? Is this problem well studied in the literature?
 A: Sometimes one can. For instance, if $H$ is an abelian subgroup: If the subgroup is cyclic, you can, if the subgroup is $Z^2$ there is a precise condition when you can (it is called the "torus theorem"). One can sometimes find reasonable conditions for this when $H$ is the fundamental group of higher genus surface.
This works better if the manifold $M$ is closed. Then whenever you have, say, a decomposition
$$
\pi_1(M)=A\star_{H} B,
$$
and $H$ is isomorphic to the fundamental group of a closed connected surface $S$, then $A, B, H$ are realized by embedded compact submanifolds.
But the best statement in general is the "virtual one":
Suppose that $H<  \pi_1(M)=G$ is a finitely generated group, $G$ is finitely generated. There is a finite-sheeted covering $p: M'\to M$ such that $H< p_*(\pi_1(M'))\le G$ and $H$ is realized as the fundamental group of a compact embedded submanifold in $M'$.
This is not always possible, but it happens surprisingly often, for instance, when $\pi_1(M)$ contains no $Z^2$-subgroups. This is related to an algebraic property called LERF.
See discussion and references in
"3-manifold groups" by Matthias Aschenbrenner, Stefan Friedl, Henry Wilton.
(a more updated version appeared as a book).
