  We consider pairs of finitely presented, residually finite groups
$P\hookrightarrow\G$ for which the induced map of profinite completions $\hat
P\to \hat\G$ is an isomorphism. We prove that there is no algorithm that, given
an arbitrary such pair, can determine whether or not $P$ is isomorphic to $\G$.
We construct pairs for which the conjugacy problem in $\G$ can be solved in
quadratic time but the conjugacy problem in $P$ is unsolvable.
  Let $\mathcal J$ be the class of super-perfect groups that have a compact
classifying space and no proper subgroups of finite index. We prove that there
does not exist an algorithm that, given a finite presentation of a group $\G$
and a guarantee that $\G\in\mathcal J$, can determine whether or not
$\G\cong\{1\}$.
  We construct a finitely presented acyclic group $\H$ and an integer $k$ such
that there is no algorithm that can determine which $k$-generator subgroups of
$\H$ are perfect.
