Decision problems and profinite completions of groups

Abstract

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