Fässler, Katrin Donne, Enrico Le 2020-04-28 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. application/pdf https://folia.unifr.ch/global/documents/308565 https://folia.unifr.ch/documents/308565/files/fae_oqi.pdf eng info:eu-repo/semantics/altIdentifier/doi/10.1007/s10711-020-00532-8 info:eu-repo/semantics/openAccess License undefined Geometriae Dedicata. - 2021, vol. 210, no. 1, p. 27-42 info:eu-repo/classification/udc/51 On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups http://purl.org/coar/resource_type/c_6501