Grothendieck's problem for 3-manifold groups (Q664241)

The problem referred to in the title of the paper is the following: Let \(u: H \rightarrow G\) be a homomorphism of finitely presented residually finite groups for which the extension \(\hat{u}: \hat{H} \rightarrow \hat{G}\) between profinite completions is an isomorphism. Is \(u\) an isomorphism? The authors introduce a notion of Grothendieck rigidity as follows: Let \(G\) be a group and \(H < G\). \((G, H)\) is a Grothendieck Pair if the inclusion of \(H\) in \(G\) provides a negative answer to Grothendieck's problem above. If for all finitely generated subgroups \(H < G\), \((G, H)\) is not a Grothendieck Pair then \(G\) is said to be Grothendieck rigid. The authors prove that any closed 3-manifold admitting a geometric structure is Grothendieck rigid. They also show that any finite volume, cusped hyperbolic 3-manifold is Grothendieck rigid.