  We show that any closed hyperbolic $3$-manifold $M$ has a co-final tower of
covers $M_i \to M$ of degrees $n_i$ such that any subgroup of $\pi_1(M_i)$
generated by $k_i$ elements is free, where $k_i \ge n_i^C$ and $C = C(M) > 0$.
Together with this result we show that $\log k_i \geq C_1 sys_1(M_i)$, where
$sys_1(M_i)$ denotes the systole of $M_i$, thus providing a large set of new
examples for a conjecture of Gromov. In the second theorem $C_1> 0$ is an
absolute constant. We also consider a generalization of these results to
non-compact finite volume hyperbolic $3$-manifolds.
