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

Sam Shepherd; Daniel J. Woodhouse

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arXiv · 2007.10034

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

Sam Shepherd·Daniel J. Woodhouse

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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.

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Sam Shepherd, Daniel J. Woodhouse. 2020-07-20. Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups. https://arxiv.org/abs/2007.10034

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