Quasi-isometries between groups with infinitely many ends

Let G, F be finitely generated groups with infinitely many ends and let π1(Γ, A), π1 (Δ, B) be graph of groups decompositions of F, G such that all edge groups are finite and all vertex groups have at most one end. We show that G, F are quasi-isometric if and only if every one-ended vertex group of π1(Δ, A) is quasi-isometric to some one-ended vertex group of π1 (Δ, B) and every one-ended vertex group of π1(Δ, B) is quasi-isometric to some one-ended vertex group of π1(Γ, 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.