Nilpotent groups and biLipschitz embeddings into $L^1$
Metric Geometry
2021-12-22 v1 Analysis of PDEs
Differential Geometry
Functional Analysis
Group Theory
Abstract
We prove that if a simply connected nilpotent Lie group quasi-isometrically embeds into an space, then it is abelian. We reach this conclusion by proving that every Carnot group that biLipschitz embeds into is abelian. Our proof follows the work of Cheeger and Kleiner, by considering the pull-back distance of a Lipschitz map into 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
Cite
@article{arxiv.2112.11402,
title = {Nilpotent groups and biLipschitz embeddings into $L^1$},
author = {Sylvester Eriksson-Bique and Chris Gartland and Enrico Le Donne and Lisa Naples and Sebastiano Nicolussi-Golo},
journal= {arXiv preprint arXiv:2112.11402},
year = {2021}
}