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 surface subgroups of closed 3-manifold groups. A consequence of our characterization is that, for any compact, orientable, irreducible and ∂-irreducible 3-manifold M with nontrivial torus decomposition, π1(M) is locally extended residually finite 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.
All Science Journal Classification (ASJC) codes
- Geometry and Topology
- 20E26 (secondary)
- 57M05 (primary)