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. - 2020
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
  • English
Classification
Mathematics
License
License undefined
Identifiers
Persistent URL
https://folia.unifr.ch/unifr/documents/308565
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