  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
$\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$.
