Separation of Relatively Quasiconvex Subgroups

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.

22 pages, 2 figures. New version has numbering matching with the published version in the Pacific Journal of Mathematics, 244 no. 2 (2010) 309--334.

References in corpus (4)

• Relative hyperbolicity and relative quasiconvexity for countable groups
• Tameness of hyperbolic 3-manifolds
• Residual finiteness, QCERF, and fillings of hyperbolic groups
• Combination of quasiconvex subgroups of relatively hyperbolic groups

Cited by in corpus (9)

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• On Quasiconvexity and Relative Hyperbolic Structures
• Algorithms detecting stability and Morseness for finitely generated groups
• Combination of fully quasiconvex subgroups and its applications
• Random walks and quasi-convexity in acylindrically hyperbolic groups
• Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups