The isomorphism problem for profinite completions of residually finite groups

2014/01/13 by Martin R. Bridson, Bridson, Martin R., Henry Wilton +1
Computer Science · Mathematics · #(57M07 #20E18 #20F10 #20F65) #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1401.2790

openalex publication_date 2014/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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