Coarse differentiation and quasi-isometries of a class of solvable Lie groups II

TL;DR

This paper extends previous work to compute the quasi-isometry group of certain solvable Lie groups, showing rigidity results and characterizing groups quasi-isometric to these Lie groups as virtually lattices in specific solvable groups.

Contribution

It provides a detailed computation of the quasi-isometry group for a subclass of solvable Lie groups and establishes quasi-isometric rigidity and classification results.

Findings

Finitely generated groups quasi-isometric to these Lie groups are polycyclic.

Such groups are virtually lattices in abelian-by-abelian solvable Lie groups.

An example of a unimodular solvable Lie group not quasi-isometric to any finitely generated group.

Abstract

In this paper, we continue with the results in \cite{Pg} and compute the group of quasi-isometries for a subclass of split solvable unimodular Lie groups. Consequently, we show that any finitely generated group quasi-isometric to a member of the subclass has to be polycyclic, and is virtually a lattice in an abelian-by-abelian solvable Lie group. We also give an example of a unimodular solvable Lie group that is not quasi-isometric to any finitely generated group, as well deduce some quasi-isometric rigidity results.