Pseudo-\(n\)-transitivity of the automorphism group of a geometric structure (Q1373239)

The author shows that local groups \(G\) of \(C^r\)-diffeomorphisms of a manifold \(M\) act pseudo-\(n\)-transitively for each \(n\geq 1\), i.e., for two \(n\)-tupels of pairwise distinct points \((x_1,\ldots,x_n)\) and \((y_1,\ldots,y_n)\) such that \(x_i\) lies in the same \(G\)-orbit as \(y_i\), and no \(G\)-orbit of dimension \(\leq 1\) contains more than one of the \(x_i\)s, there is a \(g\in G\) mapping each \(x_i\) to \(y_i\). The notion of pseudo-\(n\)-transitivity thus coincides with \(n\)-transitivity if \(G\) acts transitively on \(M\). Here a group is called local, if for all open relatively compact sets \(U\), \(V\subset M\) with \(\overline U\subset V\), each diffeotopy \(f_t\in G\) can be replaced by a diffeotopy \(g_t\in G\) that coincides for small \(t\) with \(f_t\) on \(U\) and with the identity outside \(V\). Examples of such groups include the group of leaf-preserving diffeomorphisms of a generalized foliation and the groups of Hamiltonian diffeomorphisms of Poisson, contact, and Jacobi manifolds.