**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}$.