Enrico Le Donne

Nilpotent groups, biLipschitz embeddings into L¹, and sets of finite perimeter in Carnot groups **Unusual time**

We shall present which are the nilpotent groups that admit a quasi-isometric embedding in the Banach space L¹ of integrable functions. We may consider finitely generated nilpotent groups equipped with word distances or nilpotent Lie groups equipped with left-invariant Riemannian metrics. From an asymptotic-cone argument we shall reduce to the case of bi-Lipschitz embeddings of Carnot groups. We shall prove that the only Carnot groups that embed are the abelian ones. From the work of Cheeger and Kleiner we shall see that for every Lipschitz map into L¹ one has a pullback distance obtained as a superposition of elementary distances with respect to cuts. Moreover, one only needs to consider cuts that have finite sub-Riemannian perimeter. The final goal is reached via a study of finite-perimeter sets and their blowups.
Note unusual time.