Given any infinite non-elementary hyperbolic group $G$, a theorem of Gromov asserts that there is a subgroup of $G$ isomorphic to a non-abelian free group on two generators.


Is there a similar result for a quotient of $G$? That is, is there a normal subgroup $N$ of $G$ such that $G/N$ is isomorphic to a non-abelian free group on $r>1$ generators?



Free groups as quotients of hyperbolic groups
One does not need Kazhdan property (T). Take $\mathrm{PSL}\_2(\mathbb{Z})$. The group is hyperbolic (it has a free subgroup of finite index), and is generated by an element of order 2 and an element of order 3 (it is the free product of two finite cyclic groups). Hence the generators die in every torsion-free homomorphic image. Thus $\mathrm{PSL}\_2(\mathbb{Z})$ does not have non-trivial free homomorphic images. Of course groups with property (T) do not even contain finite index subgroups that map onto free non-trivial groups (that is a much stronger property). 


