Large scale geometry of certain solvable groups

TL;DR

This paper completes the proof of quasi-isometric rigidity for a class of non-nilpotent polycyclic groups by establishing boundary rigidity of negatively curved spaces and analyzing quasi-isometries of solvable Lie groups.

Contribution

It provides the final steps in proving quasi-isometric rigidity for certain solvable groups, combining boundary rigidity results with existing work on quasi-isometries.

Findings

Proved boundary rigidity for negatively curved homogeneous spaces.

Established quasi-isometric rigidity for specific solvable groups.

Connected boundary properties with group quasi-isometries.

Abstract

In this paper we provide the final steps in the proof, announced by Eskin-Fisher-Whyte, of quasi-isometric rigidity of a class of non-nilpotent polycyclic groups. To this end, we prove a rigidity theorem on the boundaries of certain negatively curved homogeneous spaces and combine it with work of Eskin-Fisher-Whyte and Peng on the structure of quasi-isometries of certain solvable Lie groups.