Space of signatures as inverse limits of Carnot groups

TL;DR

This paper constructs a limit space of inverse systems of Carnot groups to metrize the space of signatures of rectifiable paths, revealing complex geometric structures and approximation properties related to sub-Riemannian geodesics.

Contribution

It formalizes the limit of inverse systems of metric spaces with unbounded fibers, connecting sub-Riemannian groups to the space of signatures and analyzing their geometric properties.

Findings

Limit space corresponds to the space of signatures of rectifiable paths.

The space is a geodesic metric tree with infinite valence.

Paths in Euclidean space can be approximated by projections of geodesics in Carnot groups.

Abstract

We formalize the notion of limit of an inverse system of metric spaces with $1$-Lipschitz projections having unbounded fibers. The purpose is to use sub-Riemannian groups for metrizing the space of signatures of rectifiable paths in Euclidean spaces, as introduced by Chen. The constructive limit space has the universal property in the category of pointed metric spaces with 1-Lipschitz maps. In the general setting some metric properties are discussed such as the existence of geodesics and lifts. The notion of submetry will play a crucial role. The construction is applied to the sequence of free Carnot groups of fixed rank $n$ and increasing step. In this case, such limit space is in correspondence with the space of signatures of rectifiable paths in $\mathbb R^n$. Hambly-Lyons's result on the uniqueness of signature implies that this space is a geodesic metric tree that brunches at every point with infinite valence. As a particular consequence we deduce that every path in $\mathbb R^n$ can be approximated by projections of some geodesics in some Carnot group of rank $n$, giving an evidence that the complexity of sub-Riemannian geodesics increases with the step.