Group stability and Property (T)

Abstract

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 {Gn}n=1∞, equipped with bi-invariant metrics {dn}n=1∞. We consider the case Gn=U(n) (resp. Gn=Sym(n)), equipped with the normalized Hilbert-Schmidt metric dnHS (resp. the normalized Hamming metric dnHamming). 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),dnHS) (resp. (Sym(n),dnHamming)).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(Fn), n≥3, are not stable with respect to (Sym(n),dnHamming).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),dnHS) and (Sym(n),dnHamming).