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 $nV$,