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 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.
Funding statement: The second author is grateful for the support of a Glasstone Research fellowship.
Acknowledgements
We thank the referee for helpful comments and corrections, as well as Martin Bridson, Dawid Kielak and Mark Hagen.
References
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