Nilpotent Groups and Bi-Lipschitz Embeddings Into L1
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
Published inInternational Mathematics Research Notices
DisciplineAnalyysin ja dynamiikan tutkimuksen huippuyksikköGeometrinen analyysi ja matemaattinen fysiikkaMatematiikkaAnalysis and Dynamics Research (Centre of Excellence)Geometric Analysis and Mathematical PhysicsMathematics
© The Author(s) 2022. Published by Oxford University Press.
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.
PublisherOxford University Press (OUP)
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Related funder(s)Research Council of Finland; European Commission
Funding program(s)Research costs of Academy Research Fellow, AoF; Academy Project, AoF; 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 fundingS.E.-B. was supported partially by the Finnish Academy [grant # 345005]. E.L.D. was partially supported by the Academy of Finland [grant 288501 “Geometry of Sub-Riemannian Groups” and grant 322898 “Sub-Riemannian Geometry via Metric-Geometry and Lie-Group Theory”] and by the European Research Council [ERC Starting Grant 713998 GeoMeG “Geometry of Metric Groups”]. S.N.G. was supported by the Academy of Finland [grant 328846 “Singular Integrals, Harmonic Functions, and Boundary Regularity in Heisenberg Groups”, grant 322898 “Sub-Riemannian Geometry via Metric-Geometry and Lie-Group Theory”, and grant 314172 “Quantitative Rectifiability in Euclidean and Non-Euclidean Spaces”]. ...
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