Let $G$ be a discrete, finitely generated, and amenable group. Let $H$ be a group which is quasi- isometric to $G$. Is $H$ amenable? 



Amenability as a geometric property
Yes. This is essentially an immediate consequence of the [Folner sets](http://en.wikipedia.org/wiki/F%C3%B8lner_sequence) definition of amenability. You can find references at Theorem 10.23 of the article by Ghys and de la Harpe, "[Infinite groups as geometric objects](http://perso.ens-lyon.fr/ghys/articles/infinitegroups.pdf)".


