2008/10/02 by Martin R. Bridson, Bridson, Martin R. · 1 citation
Mathematics · #20E18 #20F10 #Advanced Operator Algebra Research #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR #msc:20E18 #msc:20F10
paper · pdf · doi:10.48550/arxiv.0810.0390
arxiv created 2008/10/02 · openalex publication_date 2008/10/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider pairs of finitely presented, residually finite groups P\hookrightarrow\G for which the induced map of profinite completions P→ \G 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 \G. We construct pairs for which the conjugacy problem in \G can be solved in quadratic time but the conjugacy problem in P is unsolvable. Let \mathcal 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 \G and a guarantee that \G∈\mathcal J, can determine whether or not \G≅\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.