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

References

^ 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. S2CID 96422853.

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