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 G which are locally compact and have a continuous, faithful group action on M, it 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 Zp of p-adic integers, for some prime number p. An equivalent form of the conjecture is that Zp 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 the 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.
- Smith, Paul A. (1941). "Periodic and nearly periodic transformations". in Wilder, R.; Ayres, W. Lectures in Topology. Ann Arbor, MI: University of Michigan Press. pp. 159–190.
- Repovš, Dušan; Ščepin, Evgenij V. (June 1997). "A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps". Mathematische Annalen 308 (2): 361–364. doi:10.1007/s002080050080.
- Martin, Gaven (1999). "The Hilbert-Smith conjecture for quasiconformal actions". Electronic Research Announcements of the American Mathematical Society 5 (9): 66–70.
- Pardon, John (2013). "The Hilbert–Smith conjecture for three-manifolds". Journal of the American Mathematical Society 26 (3): 879–899. doi:10.1090/s0894-0347-2013-00766-3.
- Tao, Terence (2011), The Hilbert-Smith conjecture, https://terrytao.wordpress.com/2011/08/13/the-hilbert-smith-conjecture/ .
Original source: https://en.wikipedia.org/wiki/Hilbert–Smith conjecture. Read more