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Groups of arbitrarily poor permutation stability

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OGGW04 - Stability and probabilistic methods

We propose a quantitative notion of permutation stability for finitely generated groups. Within the class of finitely presented groups our notion is related to, but a priori distinct from, the “stability rate” introduced by Becker and Mosheiff. 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 to our groups will be very slow. 

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