On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups

Fässler, Katrin; Donne, Enrico Le

doi:10.1007/s10711-020-00532-8

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Journal article

On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups

Fässler, Katrin Department of Mathematics, University of Fribourg, Fribourg, Switzerland - Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland
Donne, Enrico Le Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland

28.04.2020

Published in:

Geometriae Dedicata. - 2021, vol. 210, no. 1, p. 27-42

English 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.

Faculty

Faculté des sciences et de médecine

Department

Département de Mathématiques

Language