In mathematics, the Hilbert–Smith conjecture is concerned with the

transformation group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the gr ...

s of manifolds; and in particular with the limitations on

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...

s ''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 In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

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 integer In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensio ...

s, for some

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

''p''. An equivalent form of the conjecture is that ''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. It is considered by some to be a better formulation of

Hilbert's fifth problem Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathem ...

, than the characterisation in the category of

s 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. In 2013,

John Pardon John Vincent Pardon (born June 1989) is an American mathematician who works on geometry and topology. He is primarily known for having solved Gromov's problem on distortion of knots, for which he was awarded the 2012 Morgan Prize. He is currentl ...

proved the three-dimensional case of the Hilbert–Smith conjecture.

References

Further reading