Decision Problems and Profinite Completions of Groups

We consider pairs of finitely presented, residually finite groups P ↪→ Γ for which the induced map of profinite completions P̂ → Γ̂ 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 Γ. We construct pairs for which the conjugacy problem in Γ 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 Γ and a guarantee that Γ ∈ J , can determine whether or not Γ ∼= {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. For Karl Gruenberg, in memoriam