Salehi-Golsefidy–Varjú super approximation conjecture

Let $G<\operatorname{SL}_n(\mathbb Z)$ be a finitely generated subgroup. Let $\mathbb G$ be the Zariski closure of $G$, and let $\mathbb G_0$ denote its identity component. The group $G$ has the super approximation property with respect to all positive integers if and only if

$$[\mathbb G_0,\mathbb G_0]=\mathbb G_0.$$

Salehi-Golsefidy–Varjú conjecture. A finitely generated subgroup $G<\operatorname{SL}_n(\mathbb Z)$ has super approximation with respect to all positive integers exactly when the identity component of its Zariski closure is perfect. This gives a proposed characterization of super approximation for arbitrary finitely generated linear groups, extending the known results for important classes of Zariski-dense subgroups; the source provides no resolution of the conjecture.