Groups with arbitrarily poor permutation stability

TL;DR

This paper introduces a new measure of permutation stability for finitely generated groups, demonstrating that some groups can have arbitrarily poor stability, affecting algorithmic efficiency.

Contribution

It defines a new notion of permutation stability, distinct from existing concepts, and constructs groups with arbitrarily bad stability properties.

Findings

Constructed finitely generated stable groups with arbitrarily poor permutation stability.

Shows that permutation stability impacts the efficiency of sample-and-substitute algorithms.

Provides a quantitative framework for analyzing permutation stability in groups.

Abstract

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.