The isomorphism problem for profinite completions of residually finite groups

TL;DR

This paper proves the non-existence of algorithms to determine isomorphism, surjectivity, or equivalence of profinite completions for pairs of finitely presented, residually finite groups, highlighting fundamental computational limitations.

Contribution

It establishes the undecidability of key problems related to profinite completions of residually finite groups, a novel result in computational group theory.

Findings

No algorithm can decide if the induced map on profinite completions is an isomorphism.

Deciding surjectivity of the map on profinite completions is undecidable.

It is impossible to algorithmically determine if two profinite completions are isomorphic.

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 $\hat{u}: \widehat{P} \to \widehat{\Gamma}$ is an isomorphism. Nor do there exist algorithms that can decide whether $\hat{u}$ is surjective, or whether $\widehat{P}$ is isomorphic to $\widehat{\Gamma}$.