Preprint

Inserted: 14 oct 2026
Last Updated: 14 oct 2026

Year: 2026

Abstract:

In any noncommutative Carnot group, for every $0≤α<1$, we show the existence of a $C^{1,α}_H$ surface that is unrectifiable with respect to the Euclidean distance. Precisely, the surface has Euclidean Hausdorff dimension higher than its topological dimension. We show that in step-2 Carnot groups the construction gives hypersurfaces of maximal Euclidean Hausdorff dimension, equal to $q−\frac{1+α}2$, where q is the topological dimension of the group. Here the range of unrectifiability is sharp, since for α=1 the surface is rectifiable in the classical sense. In step-2 Carnot groups, it is also possible to prove that the $(q−\frac12)$-dimensional Euclidean Hausdorff measure of every C1H hypersurface is zero. These results extend the seminal work by Kirchheim and Serra Cassano in the first Heisenberg group.

Tags: SingMeas