The isomorphism problem for profinite completions of finitely presented, residually finite groups

Abstract

We consider pairs of finitely presented, residually finite groups $u:P\hookrightarrow \Gamma$. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not the associated map of profinite completions $\hat{u}:\hat{P}\to \hat{\Gamma }$ is an isomorphism. Nor do there exist algorithms that can decide whether $\hat{u}$ is surjective, or whether $\hat{P}$ is isomorphic to $\hat{\Gamma }$.