It's known that subgroups of Gromov's hyperbolic groups are not necessarily hyperbolic. 
Is there any counter-example when the quotient is Abelian. More precisely, let $G$ be a Gromov's hyperbolic group and $H$ a normal subgroup of $G$ such that $G/H$ is Abelian.
Is $H$ a Gromov's hyperbolic group?



Subgroups of Gromov's hyperbolic groups
The answer is no. The case when $H$ isn't finitely generated is trivial (think of the free groups and its commutator subgroup), examples when $H$ is finitely generated are due to Rips (see the famous  [Rips construction](http://blms.oxfordjournals.org/content/14/1/45.extract) , and when $H$ is finitely presented this was found by Brady, see  [here](http://aftermath.math.ou.edu/~nbrady/papers/sub.ps) .


