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Ahlfors‐regular distances on the Heisenberg group without biLipschitz pieces
Author(s) -
Donne Enrico,
Li Sean,
Rajala Tapio
Publication year - 2017
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms.12044
Subject(s) - heisenberg group , mathematics , lipschitz continuity , carnot cycle , bounded function , group (periodic table) , space (punctuation) , measure (data warehouse) , metric space , metric (unit) , pure mathematics , besov space , mathematical analysis , interpolation space , chemistry , physics , operations management , biochemistry , functional analysis , database , computer science , gene , economics , thermodynamics , linguistics , philosophy , organic chemistry
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‐Carathéodory distance, which preserves the Ahlfors‐regularity, and such that the Carnot‐Carathéodory 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‐Carathéodory distance for which no blow‐up admits non‐trivial dilations.