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 in
Geometriae DedicataDate
2021Discipline
Analyysin ja dynamiikan tutkimuksen huippuyksikköGeometrinen analyysi ja matemaattinen fysiikkaMatematiikkaAnalysis and Dynamics Research (Centre of Excellence)Geometric Analysis and Mathematical PhysicsMathematicsCopyright
© 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.
Publisher
SpringerISSN Search the Publication Forum
0046-5755Keywords
Publication in research information system
https://converis.jyu.fi/converis/portal/detail/Publication/35662662
Metadata
Show full item recordCollections
Related funder(s)
European Commission; Academy of FinlandFunding program(s)
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 funding
Katrin 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’).

License
Related items
Showing items with similar title or keywords.
-
Nilpotent Groups and Bi-Lipschitz Embeddings Into L1
Eriksson-Bique, Sylvester; Gartland, Chris; Le Donne, Enrico; Naples, Lisa; Nicolussi Golo, Sebastiano (Oxford University Press (OUP), 2022)We prove that if a simply connected nilpotent Lie group quasi-isometrically embeds into an L1 space, then it is abelian. We reach this conclusion by proving that every Carnot group that bi-Lipschitz embeds into L1 is ... -
Metric equivalences of Heintze groups and applications to classifications in low dimension
Kivioja, Ville; Le Donne, Enrico; Nicolussi Golo, Sebastiano (Duke University Press, 2022)We approach the quasi-isometric classification questions on Lie groups by considering low dimensional cases and isometries alongside quasi-isometries. First, we present some new results related to quasi-isometries between ... -
Topics in the geometry of non-Riemannian lie groups
Nicolussi Golo, Sebastiano (University of Jyväskylä, 2017) -
Nowhere differentiable intrinsic Lipschitz graphs
Julia, Antoine; Nicolussi Golo, Sebastiano; Vittone, Davide (Wiley, 2021)We construct intrinsic Lipschitz graphs in Carnot groups with the property that, at every point, there exist infinitely many different blow-up limits, none of which is a homogeneous subgroup. This provides counterexamples ... -
Space of signatures as inverse limits of Carnot groups
Le Donne, Enrico; Züst, Roger (EDP Sciences, 2021)We formalize the notion of limit of an inverse system of metric spaces with 1-Lipschitz projections having unbounded fibers. The construction is applied to the sequence of free Carnot groups of fixed rank n and increasing ...