Space of signatures as inverse limits of Carnot groups

Abstract

We formalize the notion of limit of an inverse system of metric spaces with $1$-Lipschitz projections having unbounded fibers. The construction is applied to the sequence of free Carnot groups of fixed rank $n$ and increasing step. In this case, the limit space is in correspondence with the space of signatures of rectifiable paths in $\mathbb R^n$, as introduced by Chen. Hambly-Lyons's result on the uniqueness of signature implies that this space is a geodesic metric tree. 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.

Direct limits of infinite-dimensional Carnot groups

“…We also fix a norm on each horizontal layer for which X k = Y k = 1 and consider the corresponding Carnot-Carathéodory distance d k F on F 2,k . In the following lemma we put together several results in [DZ19].…”

“…In [MR13,MPS19], a Rademacher-type theorem has been proven when the target is a so called Banach homogeneous group, which is a Banach space equipped with a suitable non-abelian group structure. In a recent paper [DZ19], the authors study inverse limits of free nilpotent Lie groups.…”

An introduction to infinite-dimensional differential geometry

“…Now the one can argue that the metric d ∞ should correspond to the metric on the projective limit of the system (G N (R d ), 1, d N CC ) in the category of (pointed) metric spaces (with morphisms given by submetries). However, one of the main results of [LZ21] is the somewhat surprising insight that the limiting object (G ∞ , 1, d ∞ ) can not be a topological group in the topology induced by d ∞ (let us note that G ∞ = G ∞ (R d )).…”

“…They are homogeneous groups and connected and simply connected nilpotent Lie groups (i.e. Carnot groups, which are studied in sub-Riemannian geometry, cf [LZ21]…”