Subgroup separability of surface braid groups and virtual braid groups

In this paper, we give a complete characterization of subgroup separability (also known as LERF property) for surface braid groups and virtual (singular) braid groups. More specifically, for a given surface S let [Formula: see text] be the n-string pure braid group of S. We prove that if S is large then [Formula: see text] is not LERF for [Formula: see text]. If S is not large, we prove [Formula: see text] is LERF if and only if S is not the Klein bottle; [Formula: see text] is LERF if and only if S is the disk, sphere or the projective plane; [Formula: see text] is LERF if and only if S is the sphere; and [Formula: see text] is not LERF for every [Formula: see text]. We note these results also hold for the full surface braid groups [Formula: see text], since [Formula: see text] is a finite index subgroup of [Formula: see text]. Finally, we prove that the virtual and virtual singular versions of those groups are LERF if and only if [Formula: see text].