Stability and testability: equations in permutations

We initiate the study of property testing problems concerning equations in permutations. In such problems, the input consists of permutations $\sigma_{1},\dotsc,\sigma_{d}\in\text{Sym}(n)$, and one wishes to determine whether they satisfy a certain system of equations $E$, or are far from doing so. If this computational problem can be solved by querying only a small number of entries of the given permutations, we say that $E$ is testable. For example, when $d=2$ and $E$ consists of the single equation $\mathsf{XY=YX}$, this corresponds to testing whether $\sigma_{1}\sigma_{2}=\sigma_{2}\sigma_{1}$. We formulate the well-studied group-theoretic notion of stability in permutations as a testability concept, and interpret all works on stability as testability results. Furthermore, we establish a close connection between testability and group theory, and harness the power of group-theoretic notions such as amenability and property $\text{(T)}$ to produce a large family of testable equations, beyond those afforded by the study of stability, and a large family of non-testable equations. Finally, we provide a survey of results on stability from a computational perspective and describe many directions for future research.