# A question of quasi-isometric of free groups [closed]

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

## closed as off-topic by Harish Chandra Rajpoot, user91500, Antonios-Alexandros Robotis, Shailesh, choco_addictedMar 16 '16 at 4:56

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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.)