Metric equivalences of Heintze groups and applications to classifications in low dimension

TL;DR

This paper investigates the quasi-isometric classification of low-dimensional Heintze groups and solvable Lie groups, providing new results and classifications up to dimension 5, advancing understanding of their geometric equivalences.

Contribution

It introduces new results on quasi-isometries between Heintze groups and applies these to classify certain low-dimensional solvable groups up to isometry and quasi-isometry.

Findings

Complete classification up to isometry for 4-dimensional simply connected solvable groups.

Classification of polynomial growth groups in dimension 5.

New results on quasi-isometries between Heintze groups.

Abstract

We approach the quasi-isometric classification questions on Lie groups by considering low dimensional cases and isometries alongside quasi-isometries. First, we present some new results related to quasi-isometries between Heintze groups. Then we will see how these results together with the existing tools related to isometries can be applied to groups of dimension 4 and 5 in particular. Thus we take steps towards determining all the equivalence classes of groups up to isometry and quasi-isometry. We completely solve the classification up to isometry for simply connected solvable groups in dimension 4, and for the subclass of groups of polynomial growth in dimension 5.