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The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ on any connected topological $n$-manifold." It is open (afaik) for all dimensions $n > 3$.

What are some promising directions of research related to the HSC? Specially, what are some realistic (for a beginning grad student) areas of research involving group actions of finite groups?

One example related to the HSC may be trying to investigate what properties must a (hypothetical) faithful $A_p$-action satisfy. Is this direction at all promising? (by promising here I mean, can such properties be 'realistically' derivable? I don't mean if this will work towards solving the HSC.) What are some related questions you would suggest someone investigating further involving group actions from this point of view? Any relevant papers/preprints one could look at?

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  • $\begingroup$ Where did you see the $p$-adic integers written as $A_p$? That is coming from a quote, so I'd like to see the reference. $\endgroup$
    – KConrad
    Dec 3 at 7:08
  • $\begingroup$ @KConrad: This is not very common I know. Some papers, say, this one use $A_p$ instead of $\Bbb Z_p$. $\endgroup$
    – sadman-ncc
    Dec 3 at 7:15
  • $\begingroup$ I see, that paper wants to use $\mathbf Z_p$ in "topologist's notation" to mean the integers mod $p$ rather than the $p$-adic integers. $\endgroup$
    – KConrad
    Dec 3 at 7:23
  • $\begingroup$ The question seems far too broad if it allows things outside HSC (i.e., about groups action in general, even group actions on topological manifolds). $\endgroup$
    – YCor
    Dec 3 at 11:04
  • $\begingroup$ There seem to be a number of partial results on HSC in arbitrary dimension, but using weakenings of "smoothness" assumptions. Unfortunately I'm not aware of any good survey. $\endgroup$
    – YCor
    Dec 3 at 11:06

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