A characterization on separable subgroups of 3-manifold groups

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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.

Original languageEnglish (US)
Pages (from-to)187-236
Number of pages50
JournalJournal of Topology
Issue number1
StatePublished - Mar 1 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Geometry and Topology


  • 20E26 (secondary)
  • 57M05 (primary)
  • 57M50
  • 57N10

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