Persistence of freeness for Lie pseudogroup actions

The action of a Lie pseudogroup G\mathcal{G} on a smooth manifold M induces a prolonged pseudogroup action on the jet spaces J ⁿ of submanifolds of M. We prove in this paper that both the local and global freeness of the action of G\mathcal{G} on J ⁿ persist under prolongation in the jet order n. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the prolonged pseudogroup actions.