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

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 are not 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.

Automated summary

In this paper, we study the quasi-isometric rigidity of a class of groups that can be decomposed into graphs of groups with virtually free vertex groups and two-ended edge groups. Our main result states that any group quasi-isometric to a certain group G is abstractly commensurable to G, given that G is one-ended, hyperbolic relative to virtually abelian subgroups, and has a JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that are not quadratically hanging. Our result also applies to certain generic HNN extensions of a free group over cyclic subgroups.