Separation of Relatively Quasiconvex Subgroups

TL;DR

This paper investigates subgroup separability in various classes of hyperbolic and relatively hyperbolic groups, establishing new results under certain residual finiteness assumptions and using combination and filling techniques.

Contribution

It proves quasiconvex subgroup separability in relatively hyperbolic groups with nilpotent peripherals and relates LERF properties of hyperbolic 3-manifolds.

Findings

Quasiconvex subgroups are separable in certain relatively hyperbolic groups.

Geometrically finite subgroups in rank one symmetric spaces are separable.

LERF for finite volume hyperbolic 3-manifolds follows from the closed case.

Abstract

Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric spaces are separable; Kleinian groups are subgroup separable. We also show that LERF for finite volume hyperbolic 3-manifolds would follow from LERF for closed hyperbolic 3-manifolds. The method is to reduce, via combination and filling theorems, the separability of a quasiconvex subgroup of a relatively hyperbolic group G to the separability of a quasiconvex subgroup of a hyperbolic quotient G/N. A result of Agol, Groves, and Manning is then applied.