4 problems

Super approximation conjecture. The group $\Gamma$ has the super approximation property with respect to all positive integers if and only if $\mathbb G_0$ is perfect.

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 gr…

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 i…

Super-approximation conjecture. The group $\Gamma$ has super-approximation, without restriction on the moduli, if and only if the Zariski closure $\mathbb G$ of $\Gamma$ has perfec…