2
$\begingroup$

Let $G$ be a (finite dimensional real) Lie group, and let $A\subset G$ be an open neighborhood of identity. If $A=\operatorname{Exp}(\mathcal{A})$ is the injective range of the exponential map from a star-shaped neighborhood $\mathcal{A}\subset\mathcal{G}$ of the origin in the Lie algebra $\mathcal{G}$ of $G$, then $A$ is a uniquely divisible neighborhood of identity, i.e., $\exists!\sqrt[m]{x}\in A$ for all $x\in A$ and $m\in\mathbb{N}$.

Question: Given a locally compact Hausdorff topological group $G$, is it always possible to find an open uniquely divisible neighborhood of identity in $G$? If no, then what are the necessary/sufficient conditions?

If $G$ is discrete the question seems to trivialize, so let's assume $G$ is non-discrete. If $G$ is not Lie then I think it needs to have non-trivial subgroups in every neighborhood of identity, but I am not sure this is related.

Thanks in advance.

$\endgroup$
  • 1
    $\begingroup$ Your question is very false for groups like the $p$-adic numbers, the profinite completion of the integers, etc etc. $\endgroup$ – znt Jan 3 '17 at 19:52
  • $\begingroup$ Thanks. I understand that saying "question is very false" you mean that the statement in the question is far from being true (did you mean something else?). The examples you brought are totally disconnected, aren't they? I guess it is difficult to find small open subsets at all in those topologies, hence the problem. Thanks for pointing this out (again, if I have understood your assertion correctly). $\endgroup$ – Bedovlat Jan 3 '17 at 20:10
  • 1
    $\begingroup$ Heh, I guess for the $p$-adic numbers one could use the $p$-adic numbers themselves :-) so maybe the $p$-adic integers are a better example. Yes, I just mean that in my mind your question is ludicrous because if you asked me to think of a locally compact Hausdorff topological group then because of my background the first 5 examples I'd think of would not satisfy your question. $\endgroup$ – znt Jan 3 '17 at 20:10
  • $\begingroup$ @znt "unlikely" might be more useful than "ludicrous"... ? $\endgroup$ – paul garrett Jan 4 '17 at 0:01
  • 1
    $\begingroup$ @YCor: As mentioned by znt, the additive group of the $p$-adic field $\mathbb{Q}_p$ is uniquely divisible, but is not a Lie group. Does it help to add the conditions that the neighborhood can be taken arbitrarily small, and that the neighborhood is closed under taking $m$th roots? $\endgroup$ – Dave Witte Morris Jan 4 '17 at 7:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.