3 problems

Peripheral embedding conjecture. $G$ quasiisometrically embeds into a product of binary trees if and only if each peripheral subgroup $H_t$ does as well.

Upper-half-space embedding conjecture. There does not exist a quasi-isometric embedding

Let $F_2$ be the free group of rank $2$, let $F_2^n$ denote its $n$-fold direct product, and let $\mathcal{H}_g$ be the handlebody group of genus $g$. A subgroup or map is quasi-is…