Decision problems and profinite completions of groups

We consider pairs of finitely presented, residually finite groups P -> Gamma for which the induced map of profinite completions P -> Gamma 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 Gamma. We construct pairs for which the conjugacy problem in Gamma can be solved in quadratic time but the conjugacy problem in P is unsolvable. Let 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 Gamma and a guarantee that Gamma epsilon 3, can determine whether or not Gamma congruent to {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. (C) 2010 Elsevier Inc. All rights reserved.