BiLipschitz Decomposition of Lipschitz Maps between Carnot Groups

Abstract

Let f : G → H be a Lipschitz map between two Carnot groups. We show that if B is a ball of G, then there exists a subset Z ⊂ B, whose image in H under f has small Hausdorff content, such that B\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 [14], 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.

Keywords

• carnot group
• sub-riemannian geometry
• lipschitz maps
• primary 53c17
• secondary 28a78