A1 Journal article (refereed)
Nilpotent Groups and Bi-Lipschitz Embeddings Into L1 (2023)
Eriksson-Bique, S., Gartland, C., Le Donne, E., Naples, L., & Nicolussi Golo, S. (2023). Nilpotent Groups and Bi-Lipschitz Embeddings Into L1. International Mathematics Research Notices, 2023(12), 10759-10797. https://doi.org/10.1093/imrn/rnac264
JYU authors or editors
Publication details
All authors or editors: Eriksson-Bique, Sylvester; Gartland, Chris; Le Donne, Enrico; Naples, Lisa; Nicolussi Golo, Sebastiano
Journal or series: International Mathematics Research Notices
ISSN: 1073-7928
eISSN: 1687-0247
Publication year: 2023
Publication date: 22/09/2022
Volume: 2023
Issue number: 12
Pages range: 10759-10797
Publisher: Oxford University Press (OUP)
Publication country: United Kingdom
Publication language: English
DOI: https://doi.org/10.1093/imrn/rnac264
Publication open access: Openly available
Publication channel open access: Partially open access channel
Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/83706
Web address of parallel published publication (pre-print): https://arxiv.org/abs/2112.11402
Abstract
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 abelian. Our proof follows the work of Cheeger and Kleiner, by considering the pull-back distance of a Lipschitz map into L1 and representing it using a cut measure. We show that such cut measures, and the induced distances, can be blown up and the blown-up cut measure is supported on “generic” tangents of the original sets. By repeating such a blow-up procedure, one obtains a cut measure supported on half-spaces. This differentiation result then is used to prove that bi-Lipschitz embeddings can not exist in the non-abelian settings.
Keywords: differential geometry; functional analysis; group theory; Lie groups; metric spaces
Contributing organizations
Related projects
- Geometry of subRiemannian groups: regularity of finite-perimeter sets, geodesics, spheres, and isometries with applications and generalizations to biLipschitz homogenous spaces
- Le Donne, Enrico
- Research Council of Finland
- Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory
- Le Donne, Enrico
- Research Council of Finland
- GeoMeG Geometry of Metric groups
- Le Donne, Enrico
- European Commission
- Singular integrals, harmonic functions, and boundary regularity in Heisenberg groups
- Fässler, Katrin
- Research Council of Finland
Ministry reporting: Yes
Reporting Year: 2022
JUFO rating: 2
