9 problems

Let $K$ be a global field, let $S$ be a finite set of places, let $G$ be the simple algebraic group in the congruence subgroup problem, and write … Let $V^K_\infty$ denote the set…

Let $k$ be the global field, let $V$ be the set of its places, and let $G$ be the algebraic group under consideration. Write $C(S,G)$ for the congruence kernel associated with a se…

Let $\Gamma\subset G(\mathbb{Q})$ be an arithmetic group, where $G$ is a simple and simply connected algebraic group over $\mathbb{Q}$. Let $C(\Gamma)$ be the kernel of the natural…

Serre's congruence kernel conjecture. If $\mathrm{rk}_S G>1$ and $\mathrm{rk}_{K_v}G>0$ for every $v\in S\setminus V^K_{\infty}$, then $C^S(G)$ should be finite; if…

Let $k$ be a global field, let $S$ be a finite set of places, and let $G$ be a simply connected absolutely almost simple $k$-group. Let $C(G,S)$ denote the congruence kernel, and l…

Let $F_2$ be the free group of rank $2$, let $\Phi_{2,r}$ denote its free solvable quotient of derived length $r$, let $C(\Phi_{2,r})$ be the kernel of the natural map from the aut…

Serre's congruence subgroup conjecture. The kernel $C$ should be finite if $\operatorname{rk}_S G\geq 2$ and $\operatorname{rk}_{K_v}G>0$ for every nonarchimedean $v\in S$, and inf…

Let $\mathbf G$ be a simple algebraic group and let $\mathcal O$ be the ring of $S$-integers in the underlying global field. A finitely generated group $\Gamma$ has bounded generat…

Margulis–Platonov conjecture. Every homomorphism of $\Gamma$ into a finite group extends continuously to $G_\Gamma$ and factors through a homomorphism