Stability, testability and property (T)

We show that if G=⟨S|E⟩ 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≥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.