Space of signatures as inverse limits of Carnot groups

We formalize the notion of limit an inverse system metric spaces with 1-Lipschitz projections having unbounded fibers. The construction is applied to sequence free Carnot groups fixed rank n and increasing step. In this case, space in correspondence signatures rectifiable paths ℝ , as introduced by Chen. Hambly-Lyons’s result on uniqueness signature implies that a geodesic tree. As particular consequence we deduce every path can be approximated some geodesics group giving evidence complexity sub-Riemannian increases