A variant of Gromov's problem on Hölder equivalence of Carnot groups

Abstract

It is unknown if there exists a locally α-Hölder homeomorphism f:R3→H1 for any 12<α≤23, although the identity map R3→H1 is locally 12-Hölder. More generally, Gromov asked: Given k and a Carnot group G, for which α does there exist a locally α-Hölder homeomorphism f:Rk→G? Here, we equip a Carnot group G with the Carnot–Carathéodory metric. In 2014, Balogh, Hajłasz, and Wildrick considered a variant of this problem. These authors proved that if k>n, there does not exist an injective, (12+)-Hölder mapping f:Rk→Hn that is also locally Lipschitz as a mapping into R2n+1. For their proof, they use the fact that Hn is purely k-unrectifiable for k>n. In this paper, we will extend their result from the Heisenberg group to model filiform groups and Carnot groups of step at most three. We will now require that the Carnot group is purely k-unrectifiable. The main key to our proof will be showing that (12+)-Hölder maps f:Rk→G that are locally Lipschitz into Euclidean space, are weakly contact. Proving weak contactness in these two settings requires understanding the relationship between the algebraic and metric structures of the Carnot group. We will use coordinates of the first and second kind for Carnot groups.