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Separation of relatively quasiconvex subgroups. (English) Zbl 1201.20024
A subgroup $$H$$ of a group $$G$$ is called ‘seperable’ if for any $$g\in G\setminus H$$ there is a homomorphism $$\pi$$ onto a finite group such that $$\pi(g)\notin\pi(H)$$. A group is ‘residually finite’ if the trivial subgroup is seperable, and ‘subgroup seperable’ or ‘LERF’ if every finitely generated subgroup is seperable. A group is called ‘slender’ if every subgroup is finitely generated.
Theorem 1.2. Suppose that all hyperbolic groups are residually finite. If $$G$$ is a torsion free relatively hyperbolic group with peripheral structure consisting of subgroups that are LERF and slender, then any relatively quasiconvex subgroup of $$G$$ is seperable.
Corollary 1.4. Suppose that all hyperbolic groups are residually finite. Let $$G$$ be a discrete, geometrically finite subgroup of the isometry group of a rank one symmetric space (for example $$G$$ could be a lattice). All the geometrically finite subgroups of $$G$$ are seperable.
Corollary 1.5. Suppose that all hyperbolic groups are residually finite. Then all finitely generated Kleinian groups are LERF.
The authors also show that LERF for finite volume hyperbolic 3-manifolds would follow from LERF for closed hyperbolic 3-manifolds.
The results are proved by reducing, via combination and filling theorems, the seperability of a relatively quasiconvex subgroup of a relatively hyperbolic group $$G$$ to that of a quasiconvex subgroup of a hyperbolic quotient $$\overline G$$. A result of Agol, Groves and Manning is then applied.

##### MSC:
 20E26 Residual properties and generalizations; residually finite groups 20F67 Hyperbolic groups and nonpositively curved groups 20E07 Subgroup theorems; subgroup growth 57M50 General geometric structures on low-dimensional manifolds
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