A1 Journal article (refereed)
On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups (2021)

Fässler, K., & Le Donne, E. (2021). On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups. Geometriae Dedicata, 210(1), 27-42. https://doi.org/10.1007/s10711-020-00532-8

JYU authors or editors

Le Donne, Enrico

Publication details

All authors or editors: Fässler, Katrin; Le Donne, Enrico

Journal or series: Geometriae Dedicata

ISSN: 0046-5755

eISSN: 1572-9168

Publication year: 2021

Volume: 210

Issue number: 1

Pages range: 27-42

Publisher: Springer

Publication country: Netherlands

Publication language: English

DOI: https://doi.org/10.1007/s10711-020-00532-8

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/69065

Web address of parallel published publication (pre-print): https://arxiv.org/abs/1811.02253

Abstract

This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi-isometric classification with the bi-Lipschitz classification. On the other hand, we study the problem whether two quasi-isometrically equivalent Lie groups may be made isometric if equipped with suitable left-invariant Riemannian metrics. We show that this is the case for three-dimensional simply connected groups, but it is not true in general for multiply connected groups. The counterexample also demonstrates that ‘may be made isometric’ is not a transitive relation.

Keywords: group theory; geometry; differential geometry; metric spaces

Free keywords: Lie groups; quasi-isometric; bi-Lipschitz; isometric; Riemannian manifold; classification

Fields of science: