Unrectifiable surfaces with intrinsic regularity in Carnot groups

In any noncommutative Carnot group, for every $0\le \alpha <1$, we show the existence of a $C_{\mathrm{H}}^{1,\alpha }$ 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 $q-\frac{1+\alpha }{2}$, where $q$ is the topological dimension of the group. Here the range of unrectifiability is sharp, since for $\alpha =1$ the surface is rectifiable in the classical sense. In step-2 Carnot groups, it is also possible to prove that the $(q-\frac{1}{2})$-dimensional Euclidean Hausdorff measure of every $C_{\mathrm{H}}^1$ hypersurface is zero. These results extend the seminal work by Kirchheim and Serra Cassano in the first Heisenberg group.