# Engulfing Kleinian groups?

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?

• In other words, you want to know if Kleinian groups are LERF. It depends on what you exactly mean by a Kleinian group. – Misha Nov 15 '15 at 15:59
• @Misha The property I am asking about seems to be properly weaker than LERF. For me, a Kleinian group is a discrete subgroup of $\mathrm{PSL}_2(\mathbb{C})$. – Pablo Nov 15 '15 at 16:08
• See Ian Agol's example in mathoverflow.net/questions/89439/…. – Misha Nov 15 '15 at 16:19
• @Misha this is surprising for me! I am ready to assume that the group $G$ is finitely generated... – Pablo Nov 15 '15 at 16:24

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.