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

2020/07/20 by Sam O. Shepherd, Shepherd, Sam, Daniel J. Woodhouse +1 · 3 citations
Computer Science · Mathematics · #20F65 (Primary) 20E08 #57M10 (Secondary) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2007.10034

openalex publication_date 2020/07/20 · openalex created_date 2020/07/23 · openalex updated_date 2026/07/28

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.

Citations

• Croissance des boules et des géodésiques fermées dans les nilvariétés
• Quasi-isometries between groups with two-ended splittings

Cited by

• Connecting conformal dimension and Poincaré profiles
• Hierarchical hyperbolicity of hyperbolic-2-decomposable groups
• On the proofs of Leighton's Graph Covering Theorem, a notion dual to commensurability, and normal virtual retracts

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