A note on the notion of geometric rough paths

We use simple sub-Riemannian techniques to prove that every weak geometric p-rough path (a geometric p-rough path in the sense of [20]) is the limit in sup-norm of a sequence of canonically lifted smooth paths, uniformly bounded in p-variation, thus clarifying the two different definitions of a geometric p-rough path. Our proofs are sufficiently general to include the case of Holder- and modulus-type regularity. This allows us to extend a few classical results on Holder-spaces and p -variation spaces to the non-commutative setting necessary for the theory of rough paths. As an application, we give a precise description of the support of Enhanced Fractional Brownian Motion, and prove a conjecture by Ledoux et al.