In mathematics, the Hilbert–Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups "G" that can act effectively (faithfully) on a (topological) manifold "M". Restricting to groups "G" which are locally compact and have a continuous, faithful group action on "M", the conjecture states that "G" must be a Lie group.
Because of known structural results on "G", it is enough to deal with the case where "G" is the additive group formula_1 of p-adic integers, for some prime number "p". An equivalent form of the conjecture is that formula_1 has no faithful group action on a topological manifold.
The naming of the conjecture is for David Hilbert, and the American topologist Paul A. Smith. It is considered by some to be a better formulation of Hilbert's fifth problem, than the characterisation in the category of topological groups of the Lie groups often cited as a solution.
In 1997, Dušan Repovš and Evgenij Ščepin proved the Hilbert–Smith conjecture for groups acting by Lipschitz maps on a Riemannian manifold using covering, fractal, and cohomological dimension theory.
In 1999, Gaven Martin extended their dimension-theoretic argument to quasiconformal actions on a Riemannian manifold and gave applications concerning unique analytic continuation for Beltrami systems.
In 2013, John Pardon proved the three-dimensional case of the Hilbert–Smith conjecture.
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