= Solution
In the <truncated tensor algebra>, let $\mathfrak g^N(V)$ be the linear span of all iterated <Lie brackets> of elements of $V$ of length at most $N$, with bracket $[a,b]=a\otimes b-b\otimes a$. Brackets of length greater than $N$ vanish. The <free step-N nilpotent Lie group> is
$$
G^N(V)=\exp\mathfrak g^N(V)\subseteq1+\bigoplus_{k=1}^N V^{\otimes k}.
$$
The exponential and logarithm are the finite series
$$
\exp(\ell)=\sum_{j=0}^N\frac{\ell^{\otimes j}}{j!},\qquad
\log(1+a)=\sum_{j=1}^N\frac{(-1)^{j+1}a^{\otimes j}}j.
$$
Its multiplication is the tensor multiplication above, its identity is $e=(1,0,\ldots,0)$, and its inverse is
$$
(1+a)^{-1}=\sum_{j=0}^N(-a)^{\otimes j},\qquad
(\exp\ell)^{-1}=\exp(-\ell).
$$
The truncated <Baker--Campbell--Hausdorff formula> shows that this multiplication stays in the exponential of the <Lie algebra>. This is the connected simply connected <Lie group> of the free step-$N$ nilpotent <Lie algebra>, rather than a discrete free nilpotent group.
Give the first layer $V$ its usual <Euclidean norm>. An absolutely continuous curve $\gamma$ in $G^N(V)$ is horizontal if
$$
\dot\gamma_t=(L_{\gamma_t})_*\dot x_t,
$$
for an absolutely continuous control $x$ in $V$. In the <truncated tensor algebra> this equation is $\dot\gamma_t=\gamma_t\otimes\dot x_t$, truncated at degree $N$. Its horizontal length is $\int|\dot x_t|\,dt$. The <Carnot-Carathéodory distance> is
$$
\boxed{d_{CC}(g,h)=\inf_{\gamma_0=g,\,\gamma_1=h\atop\gamma\ \mathrm{horizontal}}\int_0^1|\dot x_t|\,dt.}
$$
The first-layer <Lie brackets> span the full <Lie algebra>, so horizontal curves reach every endpoint. Thus this infimum is finite; it is a left-invariant distance inducing the usual manifold topology. Dilations $\delta_\lambda(a)^{(k)}=\lambda^k a^{(k)}$ give $d_{CC}(\delta_\lambda g,\delta_\lambda h)=\lambda d_{CC}(g,h)$ for $\lambda>0$. Equivalently $d_{CC}(e,g)$ is the least length of a <bounded variation> control whose step-$N$ <path signature> is $g$.
For $p\ge1$ set $N=\lfloor p\rfloor$. A <weak geometric p-rough path> is a continuous path $\mathbf X:[0,T]\to G^N(V)$, usually based at $e$, with
$$
\boxed{\sup_D\sum_{[u,v]\in D}d_{CC}(\mathbf X_u,\mathbf X_v)^p<\infty.}
$$
Its increments $\mathbf X_{s,t}=\mathbf X_s^{-1}\otimes\mathbf X_t$ obey $\mathbf X_{s,u}=\mathbf X_{s,t}\otimes\mathbf X_{t,u}$. Equivalently their degree-$k$ tensor components have finite $p/k$-variation and satisfy the <path signature> algebraic constraints. This is a group-valued definition; being a limit of smooth <path signatures> in the same $p$-variation topology is the stronger definition of a <geometric p-rough path>.
Back to article page
