# Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions

The Hilbert-Smith conjecture states that

If $G$ is a locally compact group which acts effectively on a connected manifold as a topological transformation group then is $G$ a Lie group.

It was established for actions by diffeomorphisms by Bochner and Montgomery. Later on it was also established for (compact?) actions by Lipschitz homeomorphisms (Repovs and Shchepin) and Hölder actions with very large exponent (>dim M/ dim M+2).

I am interested if the conjecture holds for Hölder actions (with small exponents). Is it plausible these arguments can be pushed to get the conjecture for Hölder actions? Or there is a fundamental obstruction?

Also, there is a 2001 preprint "A Proof of the Hilbert-Smith Conjecture" on arxiv that claims the full conjecture. I assume it's wrong as it wasn't published, but a comment from an expert would be highly appreciated.

-
Zarathustra -- an excellent question! Maybe it's worthwhile to mention that it would suffice to prove the statement for $G$ equal the (additive) group of $p$-adic integers. –  algori Jul 20 '10 at 18:46

If you search mathscinet for Hilbert-Smith conjecture you shall find some results including something in the Hölder case [Maleshich, The Hilbert-Smith conjecture for Hölder actions. Uspekhi Mat. Nauk 52 (1997), no. 2(314), 173--174; translation in Russian Math. Surveys 52 (1997), no. 2, 407--408].

I am no expert in the subject, but I have heard that McAuley's preprint "A Proof of the Hilbert-Smith Conjecture" is incorrect. A while ago there was a conference in Istanbul, where McAuley worked at the time. The conference was devoted to studying this preprint, and in particular, Shepin, who earlier proved with Repov the HS-conjecture for Lipschitz actions, was there. If memory serves me, Shepin found that McAuley's argument actually proves something stronger to which there was a counterexample known.

-