  Let G and F be finitely generated groups with infinitely many ends and let A
and B be graph of groups decompositions of F and G such that all edge groups
are finite and all vertex groups have at most one end. We show that G and F are
quasi-isometric if and only if every one-ended vertex group of A is
quasi-isometric to some one-ended vertex group of B and every one-ended vertex
group of B is quasi-isometric to some one-ended vertex group of A. From our
proof it also follows that if G is any finitely generated group, of order at
least three, the groups: G*G, G*Z,G*G*G and G* Z/2Z are all quasi-isometric.
