[Paper Review] Quasi-isometries and rigidity of solvable groups

Alex Eskin, David Fisher+1 more|ArXiv.org|Nov 27, 2005

Geometric and Algebraic Topology18 references3 citations

TL;DR

This paper establishes quasi-isometric rigidity for non-nilpotent solvable groups, proving that any finitely generated group quasi-isometric to the 3-dimensional solvable Lie group $​\operatorname{Sol}$ is virtually a lattice in $\operatorname{Sol}$. The authors introduce a novel technique called coarse differentiation to classify self-quasi-isometries of solvable Lie groups, resolving long-standing conjectures and extending results to lamplighter groups and Diestel-Leader graphs.

ABSTRACT

In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \ltimes R^n$ where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi-isometries for $R \ltimes \R^n$ proves a conjecture made by Farb and Mosher in [FM4]. Our techniques for studying quasi-isometries extend to some other classes of groups and spaces. In particular, we characterize groups quasi-isometric to any lamplighter group, answering a question of de la Harpe [dlH]. Also, we prove that certain Diestel-Leader graphs are not quasi-isometric to any finitely generated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW],[Wo1]. We also prove that certain non-unimodular, non-hyperbolic solvable Lie groups are not quasi-isometric to finitely generated groups. The results in this paper are contributions to Gromov's program for classifying finitely generated groups up to quasi-isometry [Gr2]. We introduce a new technique for studying quasi-isometries, which we refer to as "coarse differentiation".

Motivation & Objective

• To establish quasi-isometric rigidity for non-nilpotent polycyclic groups, particularly the 3-dimensional solvable Lie group $\operatorname{Sol}$.
• To classify self-quasi-isometries of solvable Lie groups $\mathbb{R} \ltimes \mathbb{R}^n$ defined by diagonalizable matrices of determinant one with no eigenvalues on the unit circle.
• To resolve conjectures by Farb and Mosher on quasi-isometric rigidity of solvable groups and by Diestel and Leader on Diestel-Leader graphs.
• To extend the scope of quasi-isometric rigidity to lamplighter groups and non-unimodular solvable Lie groups.
• To introduce and apply the technique of coarse differentiation as a new method for studying quasi-isometries without relying on asymptotic cones.

Proposed method

• Develop a new technique called coarse differentiation to analyze quasi-isometries directly on the metric space, avoiding the use of asymptotic cones.
• Use weighted averaging over geodesics in boxes to construct a measure that is coarsely preserved under quasi-isometries, exploiting the asymmetry of the geometry.
• Leverage the coarsely preserved height function to prevent orientation reversal in boxes, ensuring rigidity of geometric structure.
• Apply the classification of self-quasi-isometries to deduce rigidity results via group actions on the boundary at infinity.
• Utilize results from the literature, including those of Tullia Dymarz and Mosher-Sageev-Whyte, to deduce quasi-isometric rigidity for lamplighter groups and Diestel-Leader graphs.
• Prove that certain non-unimodular, non-hyperbolic solvable Lie groups are not quasi-isometric to any finitely generated group, using the same coarse differentiation framework.

Experimental results

Research questions

• RQ1 Are groups quasi-isometric to $\operatorname{Sol}$ virtually lattices in $\operatorname{Sol}$?
• RQ2 Is the class of polycyclic groups quasi-isometrically rigid, i.e., is being polycyclic a geometric property?
• RQ3 Are Diestel-Leader graphs quasi-isometric to any finitely generated group?
• RQ4 Can the quasi-isometry type of lamplighter groups be characterized geometrically?
• RQ5 Do non-unimodular, non-hyperbolic solvable Lie groups admit quasi-isometric models in the finitely generated world?

Key findings

• Any group quasi-isometric to $\operatorname{Sol}$ is virtually a lattice in $\operatorname{Sol}$, establishing the first quasi-isometric rigidity result for non-nilpotent solvable groups.
• The self-quasi-isometry group of $\mathbb{R} \ltimes \mathbb{R}^n$ (with the specified matrix action) is shown to be isomorphic to the group of automorphisms preserving the associated diagonalizable action.
• The conjecture of Farb and Mosher on the quasi-isometric rigidity of solvable groups is confirmed for the class of groups $\mathbb{R} \ltimes \mathbb{R}^n$ with determinant-one, non-unit eigenvalue matrices.
• It is proven that certain Diestel-Leader graphs $DL(m,n)$ with $m \neq n$ are not quasi-isometric to any finitely generated group, verifying a conjecture of Diestel and Leader.
• The paper resolves a question of de la Harpe by characterizing groups quasi-isometric to lamplighter groups.
• Non-unimodular, non-hyperbolic solvable Lie groups are shown not to be quasi-isometric to any finitely generated group, extending the scope of rigidity results.