On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups
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
Published inGeometriae Dedicata
DisciplineAnalyysin ja dynamiikan tutkimuksen huippuyksikköGeometrinen analyysi ja matemaattinen fysiikkaMatematiikkaAnalysis and Dynamics Research (Centre of Excellence)Geometric Analysis and Mathematical PhysicsMathematics
© The Authors, 2020
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.
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Related funder(s)European Commission; Academy of Finland
Funding program(s)Research post as Academy Research Fellow, AoF
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Additional information about fundingKatrin Fässler was partially supported by the Academy of Finland (Grant 285159 ‘Sub-Riemannian manifolds from a quasiconformal viewpoint’) and by the Swiss National Science Foundation (Grant 161299 ‘Intrinsic rectifiability and mapping theory on the Heisenberg group’). Enrico Le Donne was partially supported by the Academy of Finland (Grant 288501 ‘Geometry of subRiemannian groups’) and by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’). ...
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