Quasi-isometries and rigidity of solvable groups

TL;DR

This paper advances the understanding of quasi-isometric rigidity by classifying groups quasi-isometric to certain solvable Lie groups and other spaces, confirming conjectures, and introducing a new technique called coarse differentiation.

Contribution

It proves quasi-isometric rigidity results for non-nilpotent polycyclic groups and classifies self quasi-isometries of specific solvable Lie groups, confirming a conjecture by Farb and Mosher.

Findings

Groups quasi-isometric to Sol are virtually lattices in Sol.

Self quasi-isometries of R ⋉ R^n are classified, confirming Farb and Mosher's conjecture.

Certain Diestel-Leader graphs are not quasi-isometric to finitely generated groups.

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".