TITLE: Subgroups of Gromov's hyperbolic groups
QUESTION [7 upvotes]: 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?

REPLY [8 votes]: 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 , and when $H$ is finitely presented this was found by Brady, see  here .