All finitely generated 3-manifold groups are Grothendieck rigid

TL;DR

This paper proves that all finitely generated 3-manifold groups are Grothendieck rigid, meaning their profinite completions uniquely determine the group structure, with no proper subgroup having an isomorphic profinite completion.

Contribution

It establishes the Grothendieck rigidity of all finitely generated 3-manifold groups, a significant advance in understanding their algebraic and topological properties.

Findings

Finitely generated 3-manifold groups are Grothendieck rigid.

Proper subgroups do not have isomorphic profinite completions to the whole group.

Profinite completions distinguish these groups uniquely.

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.