[Paper Review] Quasi-actions on trees: research announcement

Lee Mosher, Michah Sageev+1 more|ArXiv.org|May 22, 2000

Geometric and Algebraic Topology7 references5 citations

TL;DR

This research announcement introduces a framework for quasi-isometric rigidity of graphs of groups using coarse algebraic topology and tree rigidity, proving that groups quasi-isometric to a finite graph of coarse Poincaré duality (PD) groups with bushy Bass-Serre trees are themselves fundamental groups of graphs of groups with quasi-isometric vertex and edge groups. The key result establishes quasi-isometric rigidity and classification under dimension-homogeneous and inhomogeneous conditions, with applications to hyperbolic manifolds and abelian groups via pattern rigidity.

ABSTRACT

We develop a battery of tools for studying quasi-isometric rigidity and classification problems for splittings of groups. The techniques work best for finite graphs of groups where all edge and vertex groups are coarse PD groups. For example, if Gamma is a graph of coarse PD(n) groups for a fixed n, if the Bass-Serre tree of Gamma has infinitely many ends, and if H is a finitely generated group quasi-isometric to pi_1(Gamma), then we prove that H is the fundamental group of a graph of coarse PD(n) groups, with vertex and edge groups quasi-isometric to those of Gamma. We also have quasi-isometric rigidity theorems for graphs of coarse PD groups of nonconstant dimension, under various assumptions on the edge-to-vertex group inclusions.

Motivation & Objective

• To develop techniques for quasi-isometric rigidity of graphs of groups with coarse PD groups and bushy Bass-Serre trees.
• To classify groups quasi-isometric to fundamental groups of finite graphs of coarse PD(n) groups.
• To extend rigidity results to inhomogeneous cases, such as graphs of abelian groups with dimension-reducing edge attachments.
• To apply pattern rigidity theorems—especially Abelian and Geodesic Pattern Rigidity—to strengthen quasi-isometric classification.
• To show that quasi-isometry classes in certain graph of groups constructions are determined by projective patterns of edge space directions in vertex spaces.

Proposed method

• Use of quasi-actions on trees and tree rigidity to analyze quasi-isometric groups.
• Application of coarse algebraic topology and coarse PD group theory to control group quasi-isometries.
• Employment of the Bestvina-Feighn Combination Theorem to ensure word hyperbolicity under suitable conditions.
• Use of asymptotic cones and Rademacher's Theorem to prove Abelian Pattern Rigidity for affine foliations in Euclidean space.
• Leverage Geodesic Pattern Rigidity (Schwartz) to show weak commensurability in hyperbolic manifold amalgamations.
• Construct a quasi-isometry between new and original graphs of groups that preserves edge-to-vertex patterns up to coarse equivalence.

Experimental results

Research questions

• RQ1 Under what conditions is the quasi-isometry class of a graph of coarse PD(n) groups determined by its vertex and edge group quasi-isometry types?
• RQ2 Can quasi-isometric rigidity be established for graphs of groups with non-constant dimension vertex groups?
• RQ3 To what extent do projective patterns of edge space directions in vertex spaces serve as quasi-isometry invariants?
• RQ4 How do quasi-isometries of graphs of groups induce structure-preserving maps on the associated Bass-Serre trees and vertex spaces?
• RQ5 What is the structure of the quasi-isometry group and abstract commensurator for graphs of hyperbolic manifold groups with infinite cyclic edge groups?

Key findings

• Any finitely generated group quasi-isometric to a finite graph of coarse PD(n) groups with bushy Bass-Serre tree is itself the fundamental group of a finite graph of groups with quasi-isometric vertex and edge groups.
• For graphs of abelian groups satisfying the dimension-reducing condition (*), any quasi-isometric group is a fundamental group of a graph of virtually abelian groups.
• The projective pattern of edge space directions in vertex spaces is a quasi-isometry invariant, and Abelian Pattern Rigidity ensures that such patterns are preserved under quasi-isometries.
• In the case of graphs of hyperbolic manifold groups with infinite cyclic edge groups, the quasi-isometric group is weakly commensurable to the original group or to Z.
• The quasi-isometry group of such hyperbolic graphs of groups is finite-index in the full isometry group of the associated tree of spaces, and is isomorphic to the abstract commensurator.
• The strengthened version of Theorem 3 shows that the projective pattern of edge space directions in raft vertices is preserved under quasi-isometry, ensuring structural rigidity.