Finitely generated groups exist that are permutation stable but exhibit arbitrarily bad quantitative stability, making sample-and-substitute algorithms very slow at checking relations.
Bradford,Quantifying local embeddings into finite groups.Journal of Algebra 608 (2022), 214-238
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Groups with arbitrarily poor permutation stability
Finitely generated groups exist that are permutation stable but exhibit arbitrarily bad quantitative stability, making sample-and-substitute algorithms very slow at checking relations.