Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

Abstract We study the quasi-isometric rigidity of a large family finitely generated groups that split as graphs with virtually free vertex and two-ended edge groups. Let G be group is one-ended, hyperbolic relative to abelian subgroups, has JSJ decomposition over subgroups containing only are not quadratically hanging. Our main result any abstractly commensurable . In particular, our applies certain “generic” HNN extensions cyclic subgroups.