Transitivity of tree-like equivalence for general finite-variation paths

Prove transitivity of the tree-like equivalence relation on finite-p-variation paths in arbitrary pointed metric groups, or characterize the groups and metrics for which transitivity holds.

Background

The paper defines two pointed paths to be tree-like equivalent when their concatenation with one path reversed is tree-like. Establishing transitivity is essential for this relation to define a quotient group of paths. The authors explain that the required reduction step is known in important settings but that a proof in full generality is unavailable and may depend on the underlying group and metric.

References

A direct proof of the transitivity property $\sim$ as in \cref{def:Tree-like-equivalence} for pointed paths in full generality is not known to the authors and will in general depend on the considered group and metric in the first place.

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