Subgroup separability, knot groups and graph manifolds

Subgroup separability and virtual retractions of groups

We begin by recalling that if Γ is a group, andH a subgroup of Γ, then Γ is calledH-separable if for every g ∈ Γ\H, there is a subgroup K of finite index in Γ such that H ⊂ K but g / ∈ K. The group Γ is called subgroup separable (or LERF) if Γ is H-separable for all finitely generated subgroups H. As is well-known LERF is a powerful property in the setting of low-dimensional topology which has ...

Subgroup Separability in Residually Free Groups

We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type FP∞ are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups.

On Subgroup Separability in Hyperbolic Coxeter Groups

On Subgroup Separability in Hyperbolic Coxeter Groups

We prove that certain hyperbolic Coxeter groups are separable on their geometrically ¢nite subgroups. Mathematics Subject Classi¢cation (2000). 20H10. Key words. hyperbolic Coxeter group, subgroup separability. 1. Introduction Recall that a subgroupH of a group G is separable in G if, given any g 2 G nH, there exists a subgroup K < G of ¢nite index with H < K and g = 2K . G is called subgroup s...

Knot Groups That Are Not Subgroup Separable

This paper answers a question of Burns, Karrass and Solitar by giving examples of knot and link groups which are not subgroup-separable. For instance, it is shown that the fundamental group of the square knot complement is not subgroup separable. Let L denote the fundamental group of the link consisting of a chain of 4 circles. It is shown that L is not subgroup separable. Furthermore, it is sh...