  In this paper, we prove that all finitely generated 3-manifold groups are
Grothendieck rigid. More precisely, for any finitely generated 3-manifold group
$G$ and any finitely generated proper subgroup $H<G$, we prove that the
inclusion induced homomorphism $\widehat{i}:\widehat{H}\to \widehat{G}$ on
profinite completions is not an isomorphism.

Geometric Topology;Group Theory