Stability and Testability

We show that if $G=\langle S | E\rangle$ is a discrete group with Property (T) then $E$, as a system of equations over $S$, is not stable (under a mild condition). That is, $E$ has approximate solutions in symmetric groups $Sym(n)$, $n \geq 1$, that are far from every solution in $Sym(n)$ under the normalized Hamming metric. The same is true when $Sym(n)$ is replaced by the unitary group $U(n)$ with the normalized Hilbert--Schmidt metric. We will recall the relevant terminology, sketch the proof in a special case, and extend the instability result to show non-testability. The discussion will lead us naturally to a slightly weaker form of stability, called flexible stability, and we will survey its recent study. Based on joint works with Alex Lubotzky and Jonathan Mosehiff.