Given any infinite non-abelian 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?

Given any infinite non-abelian 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?

Given any infinite non-abelian hyperbolic group $G$, a theorem of Gromov asserts that there is 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?