Rademacher's theorem asserts that Lipschitz functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ are differentiable almost everywhere. Such a theorem may not be sharp: if $n>1$ then there exists a Lebesgue null set $N$ in $\mathbb{R}^n$ containing a point of differentiability for every Lipschitz mapping $ f: \mathbb{R}^n\rightarrow \mathbb{R}$. Such sets are called universal differentiability sets and their construction relies on the fact that existence of an (almost) maximal directional derivative implies differentiability. We will see that maximality of directional derivatives implies differentiability in all Carnot groups where the Carnot-Caratheodory distance is suitably differentiable, which include all step 2 Carnot groups (in particular the Heisenberg group). Further, one may construct a measure zero universal differentiability set in any step 2 Carnot group. Finally, we will observe that in the Engel group, a Carnot group of step 3, things can go badly wrong.

Based on joint work with Enrico Le Donne and Gareth Speight