# Higher order Pansu derivative

Given a group $(G,*)$ there is no candidate for what can be understood as a derivative of a function $$f:G\rightarrow\mathbb{R}.$$ However, for the special case of Carnot groups there is the so-called Pansu derivative and one can get surprisingly far with this notion (of course for any Lie group one can speak about the pushforward but that is not what I am interested in). As a recall, Carnot groups are particular examples of Subriemannian manifolds: by definition a Carnot group $G$ of step $n$ is a simply connected Lie group with a stratified Lie algebra $\mathfrak{g}$, i.e. $$\mathfrak{g}=V_1\oplus\cdots\oplus V_n$$ with $[V_i,V_1]=V_{i+1}$. Due to this stratification there exists a natural dilation $$\delta_{\epsilon}:G\rightarrow G$$ One says that $f$ is Pansu differentiable at $g\in G$ if the maps $$\frac{f(g*\delta_{\epsilon}(.))-f(g)}{\epsilon}$$ converge (locally uniformly) as $\epsilon\rightarrow 0$ to a homomorphism $$Df(g):(G,*)\rightarrow (\mathbb{R},+)$$ (in fact, one take any other Carnot group instead of $\mathbb{R}$ but let's keep things simple). My question is if whether this can be extended to higher derivatives. More precisely in linear spaces $U,V$ the $k$th derivative of $F:U\rightarrow V$ is a map $$D^k F:U\rightarrow L(U^{\otimes k},V)$$ but in the setting above I already do not know how to replace $U^{\otimes k}$ (e.g. what are $k$-linear maps on the direct products of $G$). If there is such a notion, can one also make sense of Taylor approximations to $f$?