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?

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

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 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.

  • $\begingroup$ Are you using some claim that f.g. Kleinian groups are virtually torsion free? Otherwise I fail to see how do you get rid of the torsion... $\endgroup$ – Pablo Nov 15 '15 at 16:39
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    $\begingroup$ @Pablo: It is not a claim, it is a theorem, also known as Selberg's Lemma (no matter how strange it sounds to call a theorem a lemma or vice versa) and it applies to all f.g. matrix groups (in zero characteristic). $\endgroup$ – Misha Nov 15 '15 at 16:45
  • $\begingroup$ Which works of Agol, Haglund and Wise show that hyperbolic 3-mainfolds are LERF? $\endgroup$ – Pablo Nov 16 '15 at 6:45
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    $\begingroup$ @Pablo, it's a consequence of Agol's solution to the virtual Haken conjecture (and also the tameness theorem of Agol and Calegari--Gabai). Details of exactly how LERF follows are given in the book '3-manifold groups' by Aschenbrenner, Friedl and me. $\endgroup$ – HJRW Nov 16 '15 at 11:36

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