Hyperbolic groups with infinitely generated commutator subgroups 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: 


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*$[G,G]$ is infinitely generated if is $G$ is free noncyclic, and of course $[G,G]$ is finitely generated when $G$ has finite abelianization.  

*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, or Morse theory considerations of Bestvina-Brady and Brady, see  here . 
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?
 A: The question to determine, given an f.g. group $G$, which homomorphisms to abelian groups (esp. cyclic groups) have a f.g. kernel, is addressed in detail in the paper: Bieri, W. Neumann, Strebel, A geometric invariant of discrete groups, Invent.math. 90, 451477 (1987). An older reference is Neumann, W.D.: Normal subgroups with infinite cyclic quotient. Math. Sci. 4, 143-148 (1979). This is not specific to the setting of hyperbolic groups (and in particular does not supersede Agol's answer). 
A: 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). 
