Quasi-isometries between groups with infinitely many ends

Let G, F be finitely generated groups with infinitely many ends and let? 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 is quasi-isometric to some one-ended vertex group of and every one-ended vertex group of is quasi-isometric to some one-ended vertex group of?. From our proof it also follows that if G is any finitely generated group, of order at least three, the groups: and are all quasi-isometric. Received: April 7, 2000; revised version: October 6, 2000