Quasi-isometries between groups with infinitely many ends

Abstract

Let G, F be finitely generated groups with infinitely many ends and let 1 (, A), 1 (, 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 1 (, A) is quasi-isometric to some one-ended vertex group of 1 (, B) and every one-ended vertex group of 1 (, B) is quasi-isometric to some one-ended vertex group of 1 (, 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.