Rough similarity of left-invariant Riemannian metrics on some Lie groups
Enrico Le Donne
University of Fribourg, Fribourg, Switzerland; University of Jyväskylä, Jyväskylä, FinlandGabriel Pallier
Université de Lille, CNRS, UMR 8524, Lille, FranceXiangdong Xie
Bowling Green State University, Bowling Green, USA
![Rough similarity of left-invariant Riemannian metrics on some Lie groups cover](/_next/image?url=https%3A%2F%2Fcontent.preview.ems.press%2Fassets%2Fpublic%2Fimages%2Fserials%2Fcover-ggd.png&w=3840&q=90)
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 map. 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–Müller–Xie, on quasi-isometries of these solvable groups.
Cite this article
Enrico Le Donne, Gabriel Pallier, Xiangdong Xie, Rough similarity of left-invariant Riemannian metrics on some Lie groups. Groups Geom. Dyn. (2024), published online first
DOI 10.4171/GGD/790