A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps

Abstract

The classical Hilbert 5th problem [14] asks whether every ( nite-dimensional) locally Euclidean topological group is necessarily a Lie group. It was solved, in the a rmative, by von Neumann [23] for compact groups in 1933, and by Gleason [11] and by Montgomery and Zippin [20] for locally compact groups in 1952. A more general version of the Hilbert 5th problem, called the Hilbert-Smith Conjecture, asserts that among all locally compact groups only Lie groups G can act e ectively on ( nite-dimensional) manifolds M (i.e. each g ∈ G\{e} moves at least one point of M) [28]. It follows from the work of Newman [24] and Smith [29] that this conjecture is equivalent to its special case when the acting group G is the group of p-adic integers Ap. In 1946 Bochner and Montgomery [3] proved the Hilbert-Smith Conjecture for groups G acting e ectively on a manifold M by di eomorphisms. A simpler, geometrical proof was obtained by Skopenkov and the authors [25] using the idea of smooth homogeneity: a compact subset K ⊂ M of a smooth manifold M is said to be smoothly ambiently homogeneous, i.e. for each x; y ∈ K there exists a di eomorphism h : (M;K; x)→ (M;K; y). It was shown that this property implies that K is a smooth submanifold of M (therefore G ∼= K is a Lie group). The proof reveals a close relationship between homogeneity and taming theory for compact subsets of Rn, which are pinched by tangent balls (the latter problem was investigated in the past by various authors [6,10,12,16,17]). See also a very interesting paper by Hahn [13]. An interesting approach to the Hilbert-Smith conjecture is via wild Cantor sets in Rn with strong homogeneity properties. Note that the Antoine necklace