Group stability and Property (T)

摘要

In recent years, there has been a considerable amount of interest in the stability of a finitely-generated group Gamma with respect to a sequence of groups {G(n)}(n=1)(infinity), equipped with biinvariant metrics {d(n)}(n=1)(infinity). 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 Gamma 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))).