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The classical formulation of Hilbert’s fifth problem asks whether topological groups that have the topological structure of a manifold, are necessarily Lie groups. This is indeed, the case, thanks to following theorem of Gleason and Montgomery-Zippin:

Theorem 1 (Hilbert’s fifth problem) Let ${G}$ be a topological group which is locally Euclidean. Then ${G}$ is isomorphic to a Lie group.

We have discussed the proof of this result, and of related results, in previous posts. There is however a generalisation of Hilbert’s fifth problem which remains open, namely the Hilbert-Smith conjecture, in which it is a space acted on by the group which has the manifold structure, rather than the group itself:

Conjecture 2 (Hilbert-Smith conjecture) Let ${G}$ be a locally compact topological group which acts continuously and faithfully (or effectively) on a connected finite-dimensional manifold ${X}$. Then ${G}$ is isomorphic to a Lie group.

Note that Conjecture 2 easily implies Theorem 1 as one can pass to the connected component ${G^\circ}$ of a locally Euclidean group (which is clearly locally compact), and then look at the action of ${G^\circ}$ on itself by left-multiplication.

The hypothesis that the action is faithful (i.e. each non-identity group element ${g \in G \backslash \{\hbox{id}\}}$ acts non-trivially on ${X}$) cannot be completely eliminated, as any group ${G}$ will have a trivial action on any space ${X}$. The requirement that ${G}$ be locally compact is similarly necessary: consider for instance the diffeomorphism group ${\hbox{Diff}(S^1)}$ of, say, the unit circle ${S^1}$, which acts on ${S^1}$ but is infinite dimensional and is not locally compact (with, say, the uniform topology). Finally, the connectedness of ${X}$ is also important: the infinite torus ${G = ({\bf R}/{\bf Z})^{\bf N}}$ (with the product topology) acts faithfully on the disconnected manifold ${X := {\bf R}/{\bf Z} \times {\bf N}}$ by the action $\displaystyle (g_n)_{n \in {\bf N}} (\theta, m) := (\theta + g_m, m).$

The conjecture in full generality remains open. However, there are a number of partial results. For instance, it was observed by Montgomery and Zippin that the conjecture is true for transitive actions, by a modification of the argument used to establish Theorem 1. This special case of the Hilbert-Smith conjecture (or more precisely, a generalisation thereof in which “finite-dimensional manifold” was replaced by “locally connected locally compact finite-dimensional”) was used in Gromov’s proof of his famous theorem on groups of polynomial growth. I record the argument of Montgomery and Zippin below the fold.

Another partial result is the reduction of the Hilbert-Smith conjecture to the ${p}$-adic case. Indeed, it is known that Conjecture 2 is equivalent to

Conjecture 3 (Hilbert-Smith conjecture for ${p}$-adic actions) It is not possible for a ${p}$-adic group ${{\bf Z}_p}$ to act continuously and effectively on a connected finite-dimensional manifold ${X}$.

The reduction to the ${p}$-adic case follows from the structural theory of locally compact groups (specifically, the Gleason-Yamabe theorem discussed in previous posts) and some results of Newman that sharply restrict the ability of periodic actions on a manifold ${X}$ to be close to the identity. I record this argument (which appears for instance in this paper of Lee) below the fold also. James on 254A, Notes 2: Complex-analyti… Pierre de la Harpe on Goursat and Furstenberg-Weiss… Jacob Wakem on Work hard Anonymous on The Hardy–Littlewood… Anonymous on The Hardy–Littlewood… Raphael on The Hardy–Littlewood… Quill on The Hardy–Littlewood… Dwight Walsh on Why global regularity for Navi… Viki Esther Chang on About Siegelvsgrh on The Hardy–Littlewood… Terence Tao on The Hardy–Littlewood… Terence Tao on The Hardy–Littlewood… Terence Tao on The Hardy–Littlewood… Yaver Gulusoy on 275A, Notes 2: Product measure… Anonymous on The Hardy–Littlewood…