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
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International Mathematics Research NoticesDate
2023Discipline
Analyysin ja dynamiikan tutkimuksen huippuyksikköGeometrinen analyysi ja matemaattinen fysiikkaMatematiikkaAnalysis and Dynamics Research (Centre of Excellence)Geometric Analysis and Mathematical PhysicsMathematicsCopyright
© 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.
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Oxford University Press (OUP)ISSN Search the Publication Forum
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https://converis.jyu.fi/converis/portal/detail/Publication/159301233
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Related funder(s)
Research Council of Finland; European CommissionFunding program(s)
Research costs of Academy Research Fellow, AoF; ERC Starting Grant; 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 funding
S.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 Acade ...
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