Nilpotent Groups and Bi-Lipschitz Embeddings Into <i>L</i>1

Abstract We prove that if a simply connected nilpotent Lie group quasi-isometrically embeds into an $L^1$ space, then it is abelian. reach this conclusion by proving every Carnot bi-Lipschitz Our proof follows the work of Cheeger and Kleiner, considering pull-back distance Lipschitz map representing using cut measure. show such measures, induced distances, can be blown up blown-up measure supported on “generic” tangents original sets. By repeating blow-up procedure, one obtains half-spaces. This differentiation result used to embeddings not exist in non-abelian settings.