Abelian Groups Are Polynomially Stable

Author:

Becker Oren1,Mosheiff Jonathan2

Affiliation:

1. Einstein Institute of Mathematics, Hebrew University, Jerusalem, Israel, 9190401

2. Department of Computer Science, Hebrew University, Jerusalem, Israel, 9190401

Abstract

Abstract In recent years, there has been a considerable amount of interest in stability of equations and their corresponding groups. Here, we initiate the systematic study of the quantitative aspect of this theory. We develop a novel method, inspired by the Ornstein–Weiss quasi-tiling technique, to prove that abelian groups are polynomially stable with respect to permutations, under the normalized Hamming metrics on the groups $\textrm{Sym}(n)$. In particular, this means that there exists $D\geq 1$ such that for $A,B\in \textrm{Sym}(n)$, if $AB$ is $\delta $-close to $BA$, then $A$ and $B$ are $\epsilon $-close to a commuting pair of permutations, where $\epsilon \leq O\left (\delta ^{1/D}\right )$. We also observe a property-testing reformulation of this result, yielding efficient testers for certain permutation properties.

Funder

European Research Council

European Union

Israel Academy of Sciences and Humanities

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Reference19 articles.

1. Almost commuting permutations are near commuting permutations;Arzhantseva;J. Funct. Anal.,2015

2. Group stability and Property (T);Becker;J. Funct. Anal.,2020

3. Stability and testability: equations in permutations;Becker

4. Stability and invariant random subgroups;Becker;Duke Math. J.,2019

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1. Testability in group theory;Israel Journal of Mathematics;2023-09

2. Testability of relations between permutations;2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS);2022-02

3. PROPERTY OF DEFECT DIMINISHING AND STABILITY;International Electronic Journal of Algebra;2022-01-17

4. C⁎-stability of discrete groups;Advances in Mathematics;2020-10

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