Group stability and Property (T)

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 $\left\{G_{n}\right\}_{n=1}^{\infty}$, equipped with bi-invariant metrics $\left\{d_{n}\right\}_{n=1}^{\infty}$. We consider the case $G_{n}=\operatorname{U}\left(n\right)$ (resp. $G_{n}=\operatorname{Sym}\left(n\right)$), equipped with the normalized Hilbert-Schmidt metric $d_{n}^{\operatorname{HS}}$ (resp. the normalized Hamming metric $d_{n}^{\operatorname{Hamming}}$). Our main result is that if $\Gamma$ is infinite, hyperlinear (resp. sofic) and has Property $\operatorname{(T)}$, then it is not stable with respect to $\left(\operatorname{U}\left(n\right),d_{n}^{\operatorname{HS}}\right)$ (resp. $\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$). This answers a question of Hadwin and Shulman regarding the stability of $\operatorname{SL}_{3}\left(\mathbb{Z}\right)$. We also deduce that the mapping class group $\operatorname{MCG}\left(g\right)$, $g\geq 3$, and $\operatorname{Aut}\left(\mathbb{F}_n\right)$, $n\geq 3$, are not stable with respect to $\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$. Our main result exhibits a difference between stability with respect to the normalized Hilbert-Schmidt metric on $\operatorname{U}\left(n\right)$ and the (unnormalized) $p$-Schatten metrics, since many groups with Property $\operatorname{(T)}$ 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 $\left(\operatorname{U}\left(n\right),d_{n}^{\operatorname{HS}}\right)$ and $\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$.