Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces

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 inThirty-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.