Higher Brin--Thompson groups are Kazhdan

We give an algebraic criterion for certain sequences $(G_n{)}_{n\in \mathbb{N}}$ of groups to have Kazhdan’s property $(T)$ eventually. As a consequence, we prove that the Brin–Thompson groups $nV$ have Kazhdan’s property $(T)$ for large $n$. The algebraic criterion provides a uniform proof that $nV$, $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$ and ${\mathrm{E}\mathrm{L}}_n(R)$ for a finitely generated associative unital ring $R$ have property $(T)$ for large $n$. In an appendix, written by Francesco Fournier-Facio, it is shown that the group $nV$ for large $n$ is neither hyperlinear, in particular not sofic, nor $\mathrm{M}\mathrm{F}$. Further, it is not Schatten $p$-approximable for any $1\le p<\mathrm{\infty }$.