Rough similarity of left-invariant Riemannian metrics on some Lie groups

TL;DR

This paper demonstrates that all left-invariant Riemannian metrics on certain solvable Lie groups, including Heintze and Sol-type groups, are roughly similar, unifying various previous results on their quasi-isometries.

Contribution

It establishes a universal rough similarity of metrics on these Lie groups, providing a common framework for existing quasi-isometry results.

Findings

All left-invariant Riemannian metrics are roughly similar via the identity.

Unified framework for quasi-isometry results on solvable groups.

Reformulation of prior results in a common setting.

Abstract

We consider Lie groups that are either Heintze groups or Sol-type groups, which generalize the three-dimensional Lie group SOL. We prove that all left-invariant Riemannian metrics on each such a Lie group are roughly similar via the identity. This allows us to reformulate in a common framework former results by Le Donne-Xie, Eskin-Fisher-Whyte, Carrasco Piaggio, and recent results of Ferragut and Kleiner-M\"uller-Xie, on quasiisometries of these solvable groups.