A finitely generated group $G$ is called LERF if every finitely generated $H \leq G$ is closed in the profinite topology on $G$ (equivalently, there is a family of finite index subgroups of $G$ intersecting in $H$). I am interested in examples of families of groups which are known to be LERF. In view of Marshall Hall's theorem for surface groups some questions come to mind:
Is every one-relator group LERF? (NO, since by YCor there are such groups which are not even residually finite).
Which finitely presented groups are LERF?
Which groups are known to be residually finite but not LERF?
Best Answer
The group $F_2 \times F_2$ is not LERF, by Allenby-Gregorac 1973, which has lots of other results in this vein.