Let $G$ be a finitely generated group and let $H$ be a subgroup of $G$. $H$ is a codimension-1 subgroup of $G$ if $C\_{G}/H$ has more than one end, where $C\_{G}$ is the Cayley graph of $G$.


Do all codimension-1 subgroups of a $3$-manifold group correspond to the fundamental group of an immersed surface in the $3$-manifold?



Codimension-1 subgroups of 3-manifold groups
Let me make some remarks on this. As far as I know, the terminology "codimension-1 subgroup" originated from a paper of Micah Sageev


*Sageev, Michah*, [**Ends of group pairs and non-positively curved cube complexes**](http://dx.doi.org/10.1112/plms/s3-71.3.585), Proc. Lond. Math. Soc., III. Ser. 71, No. 3, 585-617 (1995). [ZBL0861.20041](https://zbmath.org/?q=an:0861.20041). [MR1347406](https://mathscinet.ams.org/mathscinet-getitem?mr=1347406).


A subgroup $H< G$ of a finitely generated group $G$ is codimension-1 if the number of ends $e(G,H) > 1$. If $\Gamma$ is a [Cayley graph](https://en.wikipedia.org/wiki/Cayley_graph) for $G$, then $e(G,H)$ is [the number of ends of the graph](https://en.wikipedia.org/wiki/End_(topology)#Ends_of_graphs_and_groups) $\Gamma/H$. In this case, Sageev calls the group "semi-splittable". Then the main theorem of the paper is 


**Theorem** Suppose $G$ is a finitely generated group. Then $G$ acts essentially
on a cubing if and only if $G$ is semi-splittable. 


A *cubing* is a complete non-positively curved (or [CAT(0)](https://en.wikipedia.org/wiki/CAT(k)_space#Hadamard_spaces)) cube complex (where "cube" refers to what is more commonly known as a ["hypercube"](https://en.wikipedia.org/wiki/Hypercube)). If it is finite-dimensional, then "essential" is equivalent to the existence of an unbounded orbit. In a CAT(0) cube complex, one has "hyperplanes" which intersect each cube in a codimension-1 cube (setting one coordinate to = 0, if a cube is parameterized as $[-1,1]^n$) or the empty set. Now the stabilizer of a hyperplane in the theorem gives rise to a codimension-1 subgroup of $G$, and conversely, a codimension-1 subgroup will stabilize a hyperplane in Sageev's construction. Note that the construction of the cube complex from the subgroup is not canonical, but relies on some choices (these choices have been codified in the notion of "wall spaces"). Hopefully this helps explain the choice of terminology. 


Now, assume $G$ is the fundamental group of a connected closed irreducible 3-manifold, and assume that $G$ has a codimension-1 subgroup (in particular, $G$ is infinite). Then $G=\pi\_1(M)$, where $M$ is an aspherical 3-manifold by the [sphere theorem.](https://en.wikipedia.org/wiki/Sphere_theorem_(3-manifolds)) Corresponding to a codimension-1 subgroup $H<G$, there is a connected covering space $N\to M$ such that $\pi\_1(N)=H$. Since $e(G,H)>1$, one has that $N$ has more than one end. Hence $H\_2(N;\mathbb{Z}) \neq 0$. Choose a non-zero homology class $z\in H\_2(N)$, then we may find an embedded closed surface $\Sigma \subset N$ such that $[\Sigma]=z$ (meaning that the map induced by the inclusion of the fundamental class of $\Sigma$ into $H\_2(N)$ is $z$). If $\Sigma$ is not $\pi\_1-$injective, then we may use the [loop theorem](https://en.wikipedia.org/wiki/Loop_theorem)
to compress $\Sigma$ to get a surface of larger Euler characteristic. So by induction, we may assume that (each component of) $\Sigma$ is [incompressible](https://en.wikipedia.org/wiki/Incompressible_surface) and hence is $\pi\_1-$injective. If $\Sigma$ is not connected, we may take a component which is still homologically non-trivial. This surface cannot be a sphere because $N$ is also irreducible (again by the sphere theorem). Thus, $\Sigma$ is a $\pi\_1-$injective surface. Corollary 5.3 in Sageev's paper states a slightly weaker version of this result. 
So we may associate to $H$ a $\pi\_1$-injective surface, but in a highly non-canonical way. This argument is due to [Swarup.](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-6.3.441) Conversely, any surface subgroup of an aspherical 3-manifold group is codimension-1 (and actually may be realized by an embedding in the corresponding cover by a theorem of [Freedman-Hass-Scott](https://mathscinet.ams.org/mathscinet-getitem?mr=695910)). 


