Free subgroups of 3-manifold groups

We show that any cocompact Kleinian group $G$ has an exhaustive filtration by normal subgroups $G_i$ such that any subgroup of $G_i $ generated by $k_i$ elements is free, where $k_i > [G:G_i]^C$ and $C = C(G) > 0$. Together with this result we prove that $\log k_i > C' sys(M_i)$, where $sys(M_i)$ denotes the systole of $M_i$, thus providing a large set of new examples for a conjecture of Gromov. In the second theorem $C'> 0$ is an absolute constant. We also consider a generalization of these results to non-compact finite volume hyperbolic 3-manifolds.
In the talk, I am going to discuss the proofs of these theorems and some related open problems. This is a joint work with Cayo Doria.