Testability in group theory
Oren Becker, Alexander Lubotzky, Jonathan Mosheiff
TL;DR
We study when systems of permutation relations are testable with a constant number of queries by associating each system $E$ with a finitely presented group $\Gamma=\langle S\mid E\rangle$ and examining the solution spaces $\mathrm{Sol}_E(n)$. The core development is the BS-rigidity/Stability framework, together with two separators (Sample and Substitute and Local Statistics Matcher) that connect algorithmic testability to geometric properties of groups; a key result is that testability (via the Local Statistics Matcher) is equivalent to BS-rigidity, and stable behavior corresponds to robustness against almost-homomorphisms. The paper provides a complete amenable-group dichotomy: BS-rigidity holds for finitely generated amenable groups, while stability is equivalent to all invariant random subgroups being co-sofic; it also yields negative results for groups with Kazhdan’s property (T) or (τ), highlighting fundamental limits. By introducing flexible stability/rigidity and invariant random-subgroup formulations, the work bridges property testing of permutation relations with deep group-theoretic invariants, offering a framework for complexity questions and potential implications for soficity and non-soficity phenomena.
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 $Γ=Γ_E$ is associated and the testability of $E$ depends only on $Γ$ (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 $Γ$ is testable/BS-rigid or not.