
> 
> Let $G$ be a Kleinian group, and let $H \lneq G$ be a finitely
>  generated subgroup. Must there be a proper finite index subgroup $U$ of $G$ containing $H$ ?
> 
> 
> 


I know that this is true for Fuchsian groups and Bianchi groups. If this is unknown in general, then:


Are there any other families of Kleinian groups for which this is known?



Engulfing Kleinian groups?
First of all, fundamental groups of compact hyperbolizable 3-manifolds (with or without boundary) are LERF, this is one corollary of the work by Agol, Haglund and Wise. The LERF property is unaffected by passing to finite index subgroups. All torsion-free finitely generated discrete subgroups of $PSL(2,C)$ are fundamental groups of compact hyperbolizable 3-manifolds (follows from Thurston's work and is explained in my book "Hyperbolic Manifolds and Discrete Groups"). Therefore, all finitely generated discrete subgroups of $PSL(2,C)$ are LERF. 


As for infinitely generated discrete subgroups, they might not even contain proper finite index subgroups: See Ian Agol's example in [Non-residually finite matrix groups](https://mathoverflow.net/questions/89439/non-residually-finite-matrix-groups) and his comment in the end. I did not think about this example for a long time, but at the time when the example was posted I thought about it and concluded that Ian was correct. 


