A question of quasi-isometric of free groups

Question

For two free groups with finite ranks, are they quasi-isometric to each other?

Answer 1

You can show that $F_{nm+1}$ is a subgroup of $F_{n+1}$ of index $m$, which implies that for any $n, m$ we have that $F_{nm+1}$ is quasi-isometric to $F_{n+1}$. In particular, setting $n = 1$ gives that all free groups of finite rank at least $2$ are quasi-isometric to $F_2$ and hence to each other. ($F_1 \cong \mathbb{Z}$ is not quasi-isometric to the others.)