TITLE: Amenability as a geometric property
QUESTION [6 upvotes]: Let  $G$   be a  discrete, finitely  generated, and amenable group.  Let  $H$  be  a  group  which is quasi- isometric  to  $G$. Is  $H$ amenable?

REPLY [11 votes]: Yes. This is essentially an immediate consequence of the Folner sets 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".