8 problems

Bounded-generation conjecture. If $\Gamma$ is any noncocompact lattice in either

Uniformity conjecture. There is a function $f:\mathbb{N}\to\mathbb{N}$ such that, for any $K$, $S$, and $R$ as above,

Let $G$ be a semisimple linear algebraic group over an infinite field $k$. Write $G^+(k)$ for the subgroup generated by the subgroups $R_u(Q)(k)$, where $Q$ ranges over all minimal…

Uniform bounded generation conjecture. There is a minimal constant $K:=K(\Phi)$ proportional to ${\rm rank}(\Phi)$ and independent of $R$ and $I$ such that

Let $R$ be a principal ideal domain, let $\Phi$ be an irreducible root system of rank at least $3$, and let $I$ be a non-trivial principal ideal of $R$. For $x\in I$,…

Let $G$ be an infinite, residually finite, boundedly generated group. Its rank gradient is … where $H$ ranges over the finite-index subgroups of $G$. Abert–Jaikin-Zapirain–Nikolov'…

Let $\Gamma$ be a noncocompact irreducible arithmetic group, and let $\operatorname{rank}_{\mathbb{R}}\Gamma$ denote its real rank. A group is boundedly generated by unipotent subg…

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…