BiLipschitz decomposition of Lipschitz maps between Carnot groups

TL;DR

This paper proves that Lipschitz maps between Carnot groups can be decomposed into biLipschitz pieces outside a small Hausdorff content set, extending previous results to groups without discretizations.

Contribution

It extends biLipschitz decomposition results to all Carnot groups, removing the previous discretization restriction.

Findings

Decomposition of Lipschitz maps into biLipschitz pieces on Carnot groups.

Existence of Carnot groups without suitable discretizations.

Controlled number of biLipschitz pieces in the decomposition.

Abstract

Let $f : G \to H$ be a Lipschitz map between two Carnot groups. We show that if $B$ is ball of $G$, then there exists a subset $Z \subset B$, whose image in $H$ under $f$ has small Hausdorff content, such that $B \backslash Z$ can be decomposed into a controlled number of pieces, the restriction of $f$ on each of which is quantitatively biLipschitz. This extends a result of \cite{meyerson}, which proved the same result, but with the restriction that $G$ has an appropriate discretization. We provide an example of a Carnot group not admitting such a discretization.