Separation of relatively quasiconvex subgroups

Abstract

We show that if all hyperbolic groups are residually finite, these statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, relatively quasiconvex subgroups are separable; geometrically finite subgroups of nonuniform 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. We prove these facts by reducing, via combination and filling theorems, the separability of a relatively quasiconvex subgroup of a relatively hyperbolic group G to that of a quasiconvex subgroup of a hyperbolic quotientḠ. A result of Agol, Groves, and Manning is then applied. 1. Main results A subgroup H of a group G is called separable if for any g ∈ G \ H there is a homomorphism π onto a finite group such that π(g) ∈ π(H). A group is said to be residually finite if the trivial subgroup is separable, and subgroup separable or LERF if every finitely generated subgroup is separable. For example, Hall [1949] showed that free groups are LERF. It follows from a theorem of Mal'cev [1958] that polycyclic (and in particular finitely generated nilpotent) groups are LERF. A group is called slender if every subgroup is finitely generated. Polycyclic groups are also slender, by a result of Hirsch [1938]. Given a relatively hyperbolic group with peripheral structure consisting of LERF and slender subgroups, we study separability of relatively quasiconvex subgroups. This is connected, via filling constructions, to residual finiteness of hyperbolic groups. It is not known whether all word-hyperbolic groups are residually finite. Consequences of a positive or negative answer to this question have been explored by several authors; see for example [

Relative hyperbolicity and relative quasiconvexity for countable groups

“…Manning and Martínez-Pedroza [24] have shown that this definition is equivalent to definition (QC-3) of the present article.…”

Residual finiteness, QCERF and fillings of hyperbolic groups

“…Examples of slender LERF groups are polycyclic groups and, in particular, finitely generated nilpotent groups (see Mal 0 cev [13]). The following theorem was conjectured (without the hypothesis of torsion-freeness) in an earlier version of this paper and proved by the third author and Martínez-Pedroza [14].…”

“…Theorem 5.1 [14] Suppose that hyperbolic groups are residually finite. Let G be a torsion-free group which is hyperbolic relative to the peripheral system P D fP 1 ; : : : ; P n g. If P i is slender and LERF for each i , then relatively quasi-convex subgroups of G are separable.…”

Combination of quasiconvex subgroups of relatively hyperbolic groups

“…Fully quasiconvex subgroups appear in the work of F. Dahmani [14] where it is shown that, under some hypothesis, the combination of relatively hyperbolic groups along fully quasiconvex subgroups is a relatively hyperbolic group.This class of subgroups also appeared in the the work by J. Manning and the author [22] to prove a consequence of the hypothetical absence of non-residually finite hyperbolic groups that was conjectured by I. Agol, D. Groves and J. Manning [1].…”