Coarse differentiation and quantitative nonembeddability for Carnot groups

TL;DR

This paper establishes lower bounds on the coarse differentiability of Lipschitz maps from Carnot groups to various metric spaces, revealing fundamental nonembeddability properties and introducing new metric invariants like Markov convexity.

Contribution

It provides the first quantitative bounds on nonembeddability of Carnot groups into certain metric spaces and proves a Bourgain-type discretization theorem for these groups.

Findings

Lower bounds for coarse differentiability of Lipschitz maps from Carnot groups.

Quantitative nonembeddability results for embeddings into Alexandrov spaces, Banach spaces, and other Carnot groups.

Carnot groups possess nontrivial Markov convexity, despite not embedding into Banach spaces with this property.

Abstract

We give lower bound estimates for the macroscopic scale of coarse differentiability of Lipschitz maps from a Carnot group with the Carnot-Carath\'{e}odory metric $(G,\dcc)$ to a few different classes of metric spaces. Using this result, we derive lower bound estimates for quantitative nonembeddability of Lipschitz embeddings of $G$ into a metric space $(X,d_X)$ if $X$ is either an Alexandrov space with nonpositive or nonnegative curvature, a superreflexive Banach space, or another Carnot group that does not admit a biLipschitz homomorphic embedding of $G$. For the same targets, we can further give lower bound estimates for the biLipschitz distortion of every embedding $f : B(n) \to X$, where B(n) is the ball of radius $n$ of a finitely generated nonabelian torsion-free nilpotent group $G$. We also prove an analogue of Bourgain's discretization theorem for Carnot groups and show that Carnot groups have nontrivial Markov convexity. These give the first examples of metric spaces that have nontrivial Markov convexity but cannot biLipschitzly embed into Banach spaces of nontrivial Markov convexity.