I proposed to a master's student to work, from the exercise in Ghys-de la Harpe's book, on the proof that a finitely generated group $G$ that is quasi-isometric to $\mathbb{Z}$ is virtually $\mathbb{Z}$. However I initially had in mind the result that gives the same conclusion from the hypothesis that $G$ has linear growth.


Do you know of any simple (and elementary, in particular without assuming Gromov's theorem on polynomial growth groups) proof that a group of linear growth is quasi-isometric to $\mathbb{Z}$?



Is there a simple proof that a group of linear growth is quasi-isometric to Z?
Here are two papers dealing with this question:
Wilkie, A. J.(4-MANC); van den Dries, L.(1-IA-S)
An effective bound for groups of linear growth.
Arch. Math. (Basel) 42 (1984), no. 5, 391--396.
20F05 


and


Imrich, Wilfried(A-MNT); Seifter, Norbert(A-MNT)
A bound for groups of linear growth.
Arch. Math. (Basel) 48 (1987), no. 2, 100--104. 


the second one using ends of groups, as indicated by Sam.


