Metric Geometry of the Signature Group for p-Variation Rough Paths
Abstract
The signatures of p-rough paths form a subgroup of sufficiently high-level truncated tensor algebras, whose inverse limit is a subgroup of the full tensor algebra. For p ≥ 1, we provide a top-down description of the signature group as the inverse limit of finite-dimensional Carnot--Carathéodory geometries in the p-variation setting. We show that every compatible choice of metrics induces a topological tree structure on the inverse-limit group, under which the signature group is not a topological group. This extends the results of Enrico Le Donne and Roland Züst from bounded variation to rough paths. We also characterise the dependence of the inverse-limit groups and their metric completions on the choice of metric, identifying them with the tree-reduced path group of Horatio Boedihardjo, Xiang Geng, Terry Lyons, and Danyu Yang.
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BibTeX
@misc{medwed2026metric,
title = {{Metric Geometry of the Signature Group for $p$-Variation Rough Paths}},
author = {Felix Medwed and Sylvie Paycha and Alexander Schmeding},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10875},
eprint = {2609.10875},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10875},
}
RIS
TY - UNPB
TI - Metric Geometry of the Signature Group for p-Variation Rough Paths
AU - Felix Medwed
AU - Sylvie Paycha
AU - Alexander Schmeding
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10875
UR - https://arxiv.org/abs/2609.10875
AB - The signatures of p-rough paths form a subgroup of sufficiently high-level truncated tensor algebras, whose inverse limit is a subgroup of the full tensor algebra. For p ≥ 1, we provide a top-down description of the signature group as the inverse limit of finite-dimensional Carnot--Carathéodory geometries in the p-variation setting. We show that every compatible choice of metrics induces a topological tree structure on the inverse-limit group, under which the signature group is not a topological group. This extends the results of Enrico Le Donne and Roland Züst from bounded variation to rough paths. We also characterise the dependence of the inverse-limit groups and their metric completions on the choice of metric, identifying them with the tree-reduced path group of Horatio Boedihardjo, Xiang Geng, Terry Lyons, and Danyu Yang.
ER -