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

  • Sam Shepherd and Daniel J. Woodhouse EMAIL logo


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.


We thank the referee for helpful comments and corrections, as well as Martin Bridson, Dawid Kielak and Mark Hagen.


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Received: 2020-09-11
Revised: 2021-10-08
Published Online: 2021-12-02
Published in Print: 2022-01-01

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