I am trying to get a sense of how often the commutator subgroup $[G,G]$ of a (Gromov) hyperbolic group $G$ is infinitely generated.


**Remarks:** 


1. $[G,G]$ is infinitely generated if is $G$ is free noncyclic, and of course $[G,G]$ is finitely generated when $G$ has finite abelianization.
2. There are examples when a hyperbolic group has an finitely generated, **normal** (infinite) subgroup (of infinite index), obtained via various versions of the  [Rips constructions](http://berstein.wordpress.com/2011/02/27/the-rips-construction-i/), or Morse theory considerations of Bestvina-Brady and Brady, see  [here](http://www.math.ou.edu/~nbrady/papers/sub.ps) .


More generally, how often is the kernel of a surjection $G\to\mathbb Z$ infinitely generated (or infinitely presented)? Here is a specific:


**Question.**
Suppose the abelianization of $G$ has rank $>1$, so that there are infinitely many surjections $G\to \mathbb Z$. Is it true that there are infinitely many surjections $G\to\mathbb Z$ whose kernel is not finitely generated?



Hyperbolic groups with infinitely generated commutator subgroups
No, consider a hyperbolic 3-manifold $M$ with $b\_1(M)=2$, and all faces of the Thurston norm fibered. Then there are only finitely many surjections to $\mathbb{Z}$ with infinitely generated kernel, corresponding to the (projective classes of the) vertices of the Thurston norm ball. 


For an explicit example, consider the Whitehead link, which has all fibered faces. This is not a hyperbolic group, but one may perform orbifold Dehn filling along the longitudes to get a closed hyperbolic orbifold with this property (since the linking number is zero). 


