Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces

TL;DR

This paper demonstrates the non-minimality of the Heisenberg group in the context of metric space mappings by constructing a new distance that preserves Ahlfors-regularity and exhibits specific biLipschitz properties, answering longstanding questions.

Contribution

It introduces a novel distance on the Heisenberg group that is biLipschitz equivalent to the Carnot-Carathéodory distance but with distinct geometric properties, and extends this construction to broader Ahlfors-regular spaces.

Findings

The Heisenberg group is not minimal in looking down.

A new distance on the Heisenberg group preserves Ahlfors-regularity and is biLipschitz to the Carnot-Carathéodory distance.

Existence of a biLipschitz distance on the Heisenberg group with no nontrivial dilations in blow-ups.

Abstract

We show that the Heisenberg group is not minimal in looking down. This answers Problem 11.15 in `Fractured fractals and broken dreams' by David and Semmes, or equivalently, Question 22 and hence also Question 24 in `Thirty-three yes or no questions about mappings, measures, and metrics' by Heinonen and Semmes. The non-minimality of the Heisenberg group is shown by giving an example of an Ahlfors $4$-regular metric space $X$ having big pieces of itself such that no Lipschitz map from a subset of $X$ to the Heisenberg group has image with positive measure, and by providing a Lipschitz map from the Heisenberg group to the space $X$ having as image the whole $X$. As part of proving the above result we define a new distance on the Heisenberg group that is bounded by the Carnot-Carath\'eodory distance, that preserves the Ahlfors-regularity, and such that the Carnot-Carath\'eodory distance and the new distance are biLipschitz equivalent on no set of positive measure. This construction works more generally in any Ahlfors-regular metric space where one can make suitable shortcuts. Such spaces include for example all snowflaked Ahlfors-regular metric spaces. With the same techniques we also provide an example of a left-invariant distance on the Heisenberg group biLipschitz to the Carnot-Carath\'eodory distance for which no blow-up admits nontrivial dilations.