A characterization on separable subgroups of 3-manifold groups

Abstract: In this paper, we give a complete characterization on which finitely generated subgroups of finitely generated $3$-manifold groups are separable. Our characterization generalizes Liu's spirality character on $\pi_1$-injective immersed surface subgroups of closed $3$-manifold groups. A consequence of our characterization is that, for any compact, orientable, irreducible and $\partial$-irreducible $3$-manifold $M$ with nontrivial torus decomposition, $\pi_1(M)$ is LERF if and only if for any two adjacent pieces in the torus decomposition of $M$, at least one of them has a boundary component with genus at least $2$.