4 problems

Scaling conjecture. There is a map $\alpha:S^k\to G$ such that

Birget's conjecture. There is a nondeterministic symmetric Turing machine accepting the language of words $w$ representing $1$ in $G$ within space

Let $k$) be a number field, let $\mathbf{G}$ be an absolutely almost simple, $k$-isotropic $k$-algebraic group, let $S$ contain all archimedean places of $k$, and let $\Gamma<G_S$…

Let $G$ and $H$ be quasi-isometric groups of type $\FP_n(R)$, where $R$ is an arbitrary coefficient ring. Write $\operatorname{FV}_{R,G}^{n}$ and $\operatorname{FV}_{R,H}^{n}$ for…