Separation of relatively quasiconvex subgroups.

*(English)*Zbl 1201.20024A 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.

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.

Reviewer: Stephan Rosebrock (Karlsruhe)

##### 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 |