The isomorphism problem for profinite completions of finitely presented, residually finite groups

We consider pairs of finitely presented, residually finite groups u: P hooked right arrow Gamma We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not the associated map of profinite completions (u) over cap: (P) over cap -> (Gamma) over cap is an isomorphism. Nor do there exist algorithms that can decide whether (u) over cap is surjective, or whether (P) over cap is isomorphic to (Gamma) over cap.