Hilbert–Smith conjecture

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 ${\displaystyle \mathbb {Z} _{p}}$ of p-adic integers, for some prime number p. An equivalent form of the conjecture is that ${\displaystyle \mathbb {Z} _{p}}$ 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.^[1] 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.^[2]

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.^[3]

In 2013, John Pardon proved the three-dimensional case of the Hilbert–Smith conjecture.^[4]