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 of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let $G$ be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that aren't quadratically hanging. Our main result is that any group quasi-isometric to $G$ is abstractly commensurable to $G$. In particular, our result applies to certain "generic" HNN extensions of a free group over cyclic subgroups.

53 pages, 3 figures; v2: minor changes including to the statement of Theorem 1.3; v3: minor changes following referee's comments; to appear in Crelles

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