All finitely generated 3-manifold groups are Grothendieck rigid

Abstract

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.