TITLE: Hyperbolic groups with infinitely generated commutator subgroups
QUESTION [10 upvotes]: 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: 

$[G,G]$ 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?

REPLY [3 votes]: 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).