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Separation of Relatively Quasiconvex Subgroups
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.Comment: 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
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