The isomorphism problem for profinite completions of residually finite groups

We consider pairs of finitely presented, residually finite groups $u:P\hookrightarrow Γ$. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not the associated map of profinite completions $\hat{u}: \widehat{P} \to \widehatΓ$ is an isomorphism. Nor do there exist algorithms that can decide whether $\hat{u}$ is surjective, or whether $\widehat{P}$ is isomorphic to $\widehatΓ$.

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