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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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