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Definitions.

  • A Polish group is a topological group $G$ that is homeomorphic to a separable complete metric space.
  • A group $G$ has no small subgroups if there exists a neighborhood $U$ of the identity that contains no nontrivial subgroup of $G$.

Example. All finite dimensional Lie groups with a countable number of components are Polish groups, and it is well-known (see here) that a Lie group $G$ has no small subgroups. (Note that the proof of this fact crucially relies on the exponential map.)

I am interested in understanding when certain Polish groups will have no small subgroups. In particular, I have two questions.

Question 1. Are there examples of non-locally compact Polish groups which have no small subgroups? Edit. YCor has provided an example in the comments of such a group.

Question 2. If the answer to Question 1 is yes, are there conditions on a non-locally compact Polish group which ensure that it has no small subgroups?

Note. Due to results of Gleason and Montgomery--Zippin, a locally compact group with no small subgroups is a Lie group.

Edit. Since Question 2 is a bit too vague, I wanted to ask a follow up question, which is more precise.

Question 3. Is it true that every non-locally compact, non-abelian, pro-discrete, Polish group has no small subgroups?

Example. An example of such a group appears in the study of tempered etale covers of $K$-analytic curves of genus $g \geq 2$ where $K$ is a $p$-adic field (see Lemma 2.1.5 and Proposition 2.1.7 of Andre’s Period mappings and differential equations. From $\mathbb{C}$ to $\mathbb{C}_p$ and Theorem 2.1 of Lepage Tempered fundamental group and metric group of Mumford curve).

Thanks in advance for the help!

Jackson Morrow
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    Q1: any infinite-dimensional Banach space makes the job. – YCor Jun 13 '19 at 21:08
  • Q2: one short answer is "yes, there are." If more precisely you're asking whether there are generalizations of this result of Gleason, Montgomery-Zippin and Yamabe, to a broader setting, I don't know. One can ask whether for Polish groups, "no small subgroups" implies "locally contractible". I'd guess counterexamples are known, but this is speculation. – YCor Jun 13 '19 at 21:13
  • @YCor thank you for your comments! The phrasing of Q2 was not very precise, but yes I was asking for generalizations of the results of Gleason, Montgomery-Zippin and Yamabe. I have edited the question a bit since I am interested in a certain type of Polish group having no small subgroups. – Jackson Morrow Jun 14 '19 at 21:23
  • I don't understand the wording in Q3. Is "every" or "some" or "no" missing? – YCor Jun 14 '19 at 21:51
  • @YCor My apologies! There was a missing "every" in the question. I have corrected the statement. Thanks again! – Jackson Morrow Jun 15 '19 at 14:36
  • But obviously an infinite product of discrete groups is pro-discrete and has small subgroups (and is not locally compact unless all but finitely many are finite). – YCor Jun 15 '19 at 20:26

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