The isomorphism problem for profinite completions of residually finite groups

Martin R. Bridson and Henry Wilton

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