On the n-transitivity of the group of equivariant diffeomorphisms

Let G be a Lie group and let M be a proper smooth G-manifold. If M is connected and (M)≥ 2, the group of diffeomorphisms of M, that are isotopic to the identity through a compactly supported isotopy, acts n-transitively on M, for any n. In this paper, we prove a version of the n-transitivity result for the group of equivariant diffeomorphisms of M. As a corollary we obtain a result concerning diffeomorphisms of the orbit space $M/G$. A special case of the result for orbit spaces gives an n-transitivity result for orbifold diffeomorphisms that was earlier proved by F. Pasquotto and T. O. Rot.