Separation of relatively quasiconvex subgroups

Jason Fox Manning and Eduardo Martínez-Pedroza

Vol. 244 (2010), No. 2, 309–334

DOI: 10.2140/pjm.2010.244.309

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 quotient Ḡ. A result of Agol, Groves, and Manning is then applied.

Keywords

relatively hyperbolic group, quasiconvex subgroup, Kleinian group, residually finite, subgroup separable, LERF, QCERF, geometrically finite, combination theorem, Dehn filling

Mathematical Subject Classification 2000

Primary: 20E26, 20F67

Secondary: 57M50

Milestones

Received: 17 December 2008

Revised: 9 July 2009

Accepted: 28 July 2009

Published: 14 December 2009