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 show that the inclusion induced homomorphism $\hat{i}:\hat{H}\to \hat{G}$ on profinite completions is not an isomorphism.