group (with $d$ generators), denoted by $G^{n}\left(\mathbb{R}^{d}\right)  $, is the subgroup of $T^{n}\left(  \mathbb{R}^{d}\right)$ corresponding to the sub-Lie algebra of $A_{T}^{n}\left(\mathbb{R}^{d}\right)$ generated by the Lie bracket $[a,b]=a\otimes b-b\otimes a$. We equip it with any symmetric, sub-additive homogeneous norm $\left\Vert \cdot\right\Vert $ (cf. [@fv2010a]) which induces the left-invariant metric $d\left(  a,b\right) = {\left\lVerta^{-1} \otimes b\right\rVert}$. Given $\mathbf{x}\in\mathcal{C}\left(  [0,T];G^{n}\left(  \mathbb{R}^{d}\right) \right)$, a continuous $G^{n}\left(  \mathbb{R}^{d}\right)$-valued path, we define the increment over $\left[  s,t\right]  \subseteq\left[  0,T\right]$ by $\mathbf{x}_{s,t}:=\mathbf{x}_{s}^{-1}\otimes\mathbf{x}_{t}$, and then the $p$-variation distance is $$\begin{aligned}
 \label{pvarDist}
d_{p-var;[0,T]}(\mathbf{x},\mathbf{y}):=\sup_{\pi}\left( \sum_{i} d(\mathbf{x}_{t_{i},t_{i+1}},\mathbf{y}_{t_{i},t_{i+1}})^{p}\right)^{\frac{1}{p}},\end{aligned}$$ where the supremum runs over all partitions $\pi=\{t_{i}\}$ of $[0, T]$. We let $\left\Vert \mathbf{x}\right\Vert _{p-var;[0,T]}:=d_{p-var;[0,T]} (\mathbf{x},0)$ $,$where $0$ denotes the constant path $\mathbf{y}_{t}=e$ for all $t \in [0, T]$.

For $p \geq 1$, the *weakly geometric p-rough paths*, which we will denote by\
$\mathcal{C}^{p-var}\left([0,T];G^{\lfloor p\rfloor} \left( \mathbb{R}^{d}\right) \right) $, is the set of continuous functions $\mathbf{x}$ from $[0,T]$ onto $G^{\lfloor
p\rfloor} \left( \mathbb{R}^{d} \right)$ such that ${\left\lVert\mathbf{x}\right\rVert}_{p-var; [0, T]} < \infty$.

The simplest example of a weakly geometric $p$-rough path is as follows, given a bounded-variation path $x$ in $\mathbb{R}^{d}$, we can compute the signature of $x$ in $G^{\lfloor p\rfloor} \left( \mathbb{R}^{d} \right)$: $$\begin{aligned}
S_{\lfloor p\rfloor }(x)_{s,t}=\left( 1,\mathbf{x}_{s,t}^{1},\mathbf{x}_{s,t}^{2},\ldots ,\mathbf{x}_{s,t}^{\lfloor p\rfloor }\right),\end{aligned}$$ where $\mathbf{x}_{s,t}^{k}$ 