Nilpotent Groups and Bi-Lipschitz Embeddings Into L1
Eriksson-Bique, Sylvester; Gartland, Chris; Le Donne, Enrico; Naples, Lisa; Nicolussi Golo, Sebastiano (2022-09-22)
Oxford University Press
22.09.2022
Sylvester Eriksson-Bique, Chris Gartland, Enrico Le Donne, Lisa Naples, Sebastiano Nicolussi Golo, Nilpotent Groups and Bi-Lipschitz Embeddings Into L1, International Mathematics Research Notices, Volume 2023, Issue 12, June 2023, Pages 10759–10797, https://doi.org/10.1093/imrn/rnac264
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© The Author(s) 2022. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
https://creativecommons.org/licenses/by/4.0/
© The Author(s) 2022. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
https://creativecommons.org/licenses/by/4.0/
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:oulu-202312193941
https://urn.fi/URN:NBN:fi:oulu-202312193941
Tiivistelmä
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.
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.
Kokoelmat
- Avoin saatavuus [37887]
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