Groups, Geometry, and Dynamics


Full-Text PDF (151 KB) | Metadata | Table of Contents | GGD summary
Volume 8, Issue 3, 2014, pp. 733–745
DOI: 10.4171/GGD/245

Published online: 2014-10-02

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

Martin R. Bridson[1] and Henry Wilton[2]

(1) University of Oxford, UK
(2) University of Cambridge, Great Britain

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

Keywords: Profinite completions, residually finite groups, finitely presented groups, decision problems

Bridson Martin, Wilton Henry: The isomorphism problem for profinite completions of finitely presented, residually finite groups. Groups Geom. Dyn. 8 (2014), 733-745. doi: 10.4171/GGD/245