Group stability and Property (T)

In recent years, there has been a considerable amount of interest in the stability of a finitely-generated group Γ with respect to a sequence of groups {G_n}_n=1 ^∞, equipped with bi-invariant metrics {d_n}_n=1 ^∞. We consider the case G_n=U(n) (resp. G_n=Sym(n)), equipped with the normalized Hilbert-Schmidt metric d_n ^HS (resp. the normalized Hamming metric d_n ^Hamming). Our main result is that if Γ is infinite, hyperlinear (resp. sofic) and has Property (T), then it is not stable with respect to (U(n),d_n ^HS) (resp. (Sym(n),d_n ^Hamming)). This answers a question of Hadwin and Shulman regarding the stability of SL₃(Z). We also deduce that the mapping class group MCG(g), g≥3, and Aut(F_n), n≥3, are not stable with respect to (Sym(n),d_n ^Hamming). Our main result exhibits a difference between stability with respect to the normalized Hilbert-Schmidt metric on U(n) and the (unnormalized) p-Schatten metrics, since many groups with Property (T) are stable with respect to the latter metrics, as shown by De Chiffre-Glebsky-Lubotzky-Thom and Lubotzky-Oppenheim. We suggest a more flexible notion of stability that may repair this deficiency of stability with respect to (U(n),d_n ^HS) and (Sym(n),d_n ^Hamming).