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 attracted a good deal of attention (see [3] and [35] for example), however, it is a property established either positively or negatively for very few classes of groups. Much recent work has suggested that the correct condition to impose on the subgroup is not finite generation but geometrical finiteness (or quasi-convexity in the case of a negatively curved group) and this article takes this theme further by exploring a new related condition which is even more relevant for topological applications and which is reminiscent of old and very classical topological considerations, namely map extension properties and ANR’s. We begin with a simple definition that underpins much of what follows.