# 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 *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 *Z _{p}* of p-adic integers, for some prime number

*p*. An equivalent form of the conjecture is that

*Z*has no faithful group action on a topological manifold.

_{p}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 the 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]}

## References[edit]

**^**Smith, Paul A. (1941). "Periodic and nearly periodic transformations". In Wilder, R.; Ayres, W (eds.).*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. arXiv:1112.2324. doi:10.1090/s0894-0347-2013-00766-3.

## Further reading[edit]

- Tao, Terence (2011),
*The Hilbert-Smith conjecture*.