On Locally Lipschitz Locally Compact Transformation Groups of Manifolds

In this paper we show that a “locally Lipschitz” locally compact transformation group acting continuously and effectively on a connected paracompact locally Euclidean topological manifold is a Lie group. This is a contribution to the proof of the Hilbert-Smith conjecture. It generalizes the classical Bochner-Montgomery-Kuranishi Theorem[1, 9] and also the Repovš-Ščepin Theorem [17] which holds only for Riemannian manifolds. A topological group G acting continuously and effectively (i.e. every non-trivial element of G acts non-trivially) on an n-dimensional topological manifold M is said to be locally Lipschitz if ∀ a ∈ M , ∃ (Ua, φa) chart of M where a ∈ Ua open ⊆ M , φa : Ua → R n an open embedding and 1 ∈ V open ⊆ G, a ∈ U open ⊆ Ua such that V (U) ⊆ Ua and ∀ g ∈ V , d(gx, gy) ≤ cgd(x, y) ∀ x, y ∈ U for some cg ∈ R where d is the transported Euclidean distance from R n via the map φa. This definition generalizes the corresponding one given in [8]. In this paper we generalize the results of [8] and we show that a locally Lipschitz locally compact transformation group acting continuously and effectively on a connected paracompact locally Euclidean manifold is a Lie group, Theorem 2. This provides a contribution to the proof of the Hilbert-Smith conjecture. It generalizes the classical Bochner-Montgomery-Kuranishi Theorem [1, 9] where the group acts by C diffeomorphisms on a smooth manifold and also the Repovš-Ščepin Theorem [17] which holds only for Riemannian manifolds. The following theorem is all what we need to establish our result. It replaces the Hausdorff dimension argument in [17] by an elementary theorem in dimension theorem from Nagata book [16, p.83]. This will enable us to obtain the required generalization from Riemannian manifolds to topological manifolds. 2000 Mathematics Subject Classification : Primary 57S05, Secondary 54H15.