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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?

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    $\begingroup$ The answer is no. The case when H isn't f.g. is trivial, examples when H is f.g is due to Rips, and when H is f.p. this was found by Brady, see aftermath.math.ou.edu/~nbrady/papers/sub.ps and references therein. $\endgroup$ – Igor Belegradek Mar 14 '14 at 18:56
  • $\begingroup$ @IgorBelegradek, why don't you post your comment as an answer? $\endgroup$ – HJRW Mar 17 '14 at 9:55
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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 .

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