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My question is in some sense a less ambitious version of the following MO question where the answer was inconclusive. A Carnot group of step $N$ can be identified within the tensor algebra, modulo ...

Let $f:[0,T]\rightarrow \mathbb{R}^d$ be a p-geometric rough path and let $\mathcal{G}_p^d$ be the collection of all such paths. Does the Lyons signature map define a continuous bijection between $\...