Testability in group theory

TL;DR

This paper explores the testability of properties related to groups defined by relations between permutations, introducing new concepts like testable groups and providing tools to determine their testability.

Contribution

It extends the study of property testing to group theory, establishing a connection between testability and group properties, and introduces methods to identify testable and BS-rigid groups.

Findings

Introduction of testable groups and BS-rigid groups

Development of tools to assess group testability

Connection between property testing and group solvability

Abstract

This paper is a journal counterpart to our FOCS 2021 paper, in which we initiate the study of property testing problems concerning a finite system of relations $E$ between permutations, generalizing the study of stability in permutations. To every such system $E$, a group $\Gamma=\Gamma_E$ is associated and the testability of $E$ depends only on $\Gamma$ (just like in Galois theory, where the solvability of a polynomial is determined by the solvability of the associated group). This leads to the notion of testable groups, and, more generally, Benjamini-Schramm rigid groups. The paper presents an ensemble of tools to check if a given group $\Gamma$ is testable/BS-rigid or not.