Equality of exponential-decay completions for Carnot–Carathéodory metrics

Determine whether the completion of the group-like elements defined using the rough-path max metrics $\rho^1=(\rho^1_k)_{k\in\mathbb N}$ coincides with the completion defined using the finite-level Carnot–Carathéodory metrics $d^{1-}=(d_k^{1-})_{k\in\mathbb N}$, where $d_k^{1-}$ is the minimum Euclidean-length cost of horizontal paths realizing a $k$-level signature.

Background

The paper associates different inverse-limit and completion constructions to different compatible families of metrics on truncated free nilpotent groups. For p=1p=1, one family is given by the max-type rough-path metrics ρk1\rho^1_k, while another is given by the intrinsic Carnot–Carathéodory distances dk1−d_k^{1-}. The authors explicitly leave unresolved whether these two choices produce the same group-like completion.

References

However, whether $G_{\rho1\text{-}p.r.c.}(V)$ with $\rho1=(\rho1_k)_{k \in N}$ coincides with $ G_{d{1-}-p.r.c.}(V)$ where $d{1-} = (d_k{1-})_{k \in N}$ is given by \begin{equation} \label{eq:CC-distance-p=1} d{1-}_{k}(1,g) := \min_{X \in \mathcal{A}{}_k(g)} \int_01 |X_t{-1}\dot X_t| \mathrm{d}t \end{equation} is an open question.

— Metric Geometry of the Signature Group for $p$-Variation Rough Paths  (2609.10875 - Medwed et al., 9 Sep 2026) in Section 2, paragraph “Lifts into the group-like elements”