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 <math>\Z_p</math> of p-adic integers, for some prime number p. An equivalent form of the conjecture is that <math>\Z_p</math> 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.<ref name=Smith1941>Template:Cite book</ref> 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.<ref name=Repovs1997>Template:Cite journal</ref>
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.<ref name=Gaven1999> Template:Cite journal</ref>
In 2013, John Pardon proved the three-dimensional case of the Hilbert–Smith conjecture.<ref name=pardon2013> Template:Cite journal</ref>
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