Summary: Subgroup separability and virtual retractions of groups
D. D. Long
, & A. W. Reid
August 25, 2004
W begin by recalling that if is a group, and H a subgroup of , then is called H-separable if for
every g \H, there is a subgroup K of finite index in such that H K but g / K. The group
is called subgroup separable (or LERF) if 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  and  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.
Definition 1.1 Let be a group and H a subgroup. Then a homomorphism : H - A extends
over the finite index subgroup V if
· H V
· There is a homomorphism : V - A which is the homomorphism when restricted to H.