Gorodnik–Varjú's super approximation conjecture

Let $G$ be a finitely generated subgroup of $\mathrm{SL}_n(\mathbb{Z})$. Write $\mathbb{G}$ for the Zariski closure of $G$, and let $\mathbb{G}_0$ be its identity component. The group $G$ has the super approximation property with respect to all positive integers if there exists $\epsilon>0$ such that the Cheeger constants of the Cayley graphs of the reductions of $G$ are greater than $\epsilon$ for every positive integer modulus. Gorodnik–Varjú's super approximation conjecture. The group $G$ has the super approximation property with respect to all positive integers if and only if $\mathbb{G}_0$ is perfect, that is,

$$[\mathbb{G}_0,\mathbb{G}_0]=\mathbb{G}_0.$$

This conjecture characterizes uniform expansion over all congruence quotients in terms of the algebraic structure of the identity component of the Zariski closure. The source attributes it to Gorodnik and Varjú; its resolution status is not specified in the supplied text.