Grothendieck's Problem for 3-Manifold Groups

The following problem was posed by Grothendieck: Let $G$ be a finitely presented residually finite group and $u : H \to G$ the inclusion homomorphism of a finitely presented subgroup. Suppose that the extension $\hat u$ to the profinite completion is an isomorphism. Is $u$ an isomorphism? This talk will discuss this problem in the context of the fundamental groups of compact 3-manifolds.