Groups with arbitrarily poor permutation stability

We propose a quantitative notion of permutation stability for finitely generated groups. Our notion is related to, but distinct from, the stability rate'' introduced by Becker and Mosheiff (which is valid within the class of finitely presented groups). We construct a family of finitely generated stable groups which exhibit, quantitatively, arbitrarily bad’’ permutation stability. This means that any application of a ``sample-and-substitute’’ algorithm will be very slow in ascertaining whether a given tuple of permutations satisfy the defining relations of our groups.