[Paper Review] Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

Sam O. Shepherd, Daniel J. Woodhouse|arXiv (Cornell University)|Jul 20, 2020

Geometric and Algebraic Topology44 references4 citations

TL;DR

This paper establishes quasi-isometric rigidity for a broad class of groups that split as graphs of virtually free groups with two-ended edge groups. Using Bass-Serre theory, JSJ decompositions, and a novel application of Leighton's theorem to graphs with colored fins, the authors prove that any group quasi-isometric to such a group is abstractly commensurable to it—extending rigidity results to generic HNN extensions and amalgamations of free groups over cyclic subgroups with rigid line patterns.

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.

Motivation & Objective

• To establish quasi-isometric rigidity for a large class of finitely generated groups that split as graphs of virtually free groups with two-ended edge groups.
• To prove that any group quasi-isometric to such a group is abstractly commensurable to it, under mild conditions including one-endedness and absence of quadratically hanging vertex groups.
• To generalize known rigidity results to generic HNN extensions and amalgamations of free groups over cyclic subgroups with rigid line patterns.
• To develop a new method combining Leighton’s theorem with colored fin structures to construct finite common covers of graphs of spaces.
• To resolve the failure of quasi-isometric rigidity in the presence of higher-rank cylindrical factors by constructing a counterexample.

Proposed method

• Employing Bass-Serre theory to analyze group actions on trees and decompose groups into graphs of groups with virtually free vertex groups and two-ended edge groups.
• Using JSJ decompositions over two-ended subgroups to identify canonical splittings and eliminate non-rigid structures such as quadratically hanging vertices.
• Introducing a new version of Leighton’s theorem for graphs with colored fins to construct finite common covers of vertex and edge spaces.
• Defining cylinder numbers, stretch ratios, and density coefficients to control the geometry of graphs of spaces and ensure quasi-isometric equivalence.
• Constructing a template for a common finite cover by analyzing link maps and local isomorphisms between vertex and edge space decompositions.
• Using a counterexample involving higher-rank cylindrical factors to show that the main rigidity result fails when the cylindrical structure exceeds rank one.

Experimental results

Research questions

• RQ1 Under what conditions is a group quasi-isometric to a graph of virtually free groups with two-ended edge groups abstractly commensurable to it?
• RQ2 Can quasi-isometric rigidity be established for generic HNN extensions of free groups over cyclic subgroups with rigid line patterns?
• RQ3 What role do JSJ decompositions and the absence of quadratically hanging vertex groups play in ensuring rigidity?
• RQ4 How can Leighton’s theorem be adapted to graphs with colored fins to construct finite common covers in geometric group theory?
• RQ5 Why does the quasi-isometric rigidity result fail for groups with higher-rank cylindrical factors, and what structural obstruction arises?

Key findings

• Any group quasi-isometric to a one-ended, relatively hyperbolic group with JSJ decomposition over two-ended subgroups and only virtually free non-quadratically hanging vertex groups is abstractly commensurable to the original group.
• The result applies to generic HNN extensions and amalgamations of free groups over cyclic subgroups, provided the associated line patterns on the vertex groups are rigid.
• A new version of Leighton’s theorem for graphs with colored fins enables the construction of finite common covers, which is central to proving commensurability.
• The degree of a vertex in the common cover is determined by the index of the image of the vertex group in the free factor, which depends on the rank of the free image and the ambient free group.
• A counterexample is constructed to show that the rigidity result fails when cylindrical factors have rank greater than one, due to a mismatch in index computations between different free group ranks.
• The proof relies on the uniqueness of tree of cylinders decompositions and the preservation of local structure under quasi-isometries, leading to a global commensurability.