  We prove that if a simply connected nilpotent Lie group quasi-isometrically
embeds into an $L^1$ space, then it is abelian. We reach this conclusion by
proving that every Carnot group that biLipschitz embeds into $L^1$ is abelian.
  Our proof follows the work of Cheeger and Kleiner, by considering the
pull-back distance of a Lipschitz map into $L^1$ 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.
