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Say a topological group $G$ is topologically characteristically simple if there does not exist a closed subgroup $1 < K < G$ such that $K$ is invariant under all automorphisms of $G$ (here `automorphisms' are required to preserve the topology as well as the group structure).

Q1: Has someone written down a classification of locally compact second-countable abelian groups that are topologically characteristically simple?

I tried to derive a classification myself; the case I am having difficulty with is when $G$ is torsion-free and the group $pG$ of $p$-th powers in $G$ is a proper dense subgroup of $G$. (Another possible division of this case is between those $G$ with a dense divisible subgroup, and those with no divisible subgroup, but I don't know where to go from there.)

I know one such group for each $p$: let $G$ be the group of all functions from $\mathbb{N}$ to $\mathbb{Q}_p$ under pointwise addition such that all but finitely many values are in $\mathbb{Z}_p$, and topologise it so that the group of functions from $\mathbb{N}$ to $\mathbb{Z}_p$ is an open subgroup with the compact-open topology. This example is topologically characteristically simple with a dense divisible subgroup (the finitely supported functions).

Q2: Are there any other locally compact second-countable abelian groups $G$ such that $G$ is torsion-free, $pG$ is a proper dense subgroup (edit: and $x^{p^n} \rightarrow 1$ for all $x \in G$)? If so, are any of them topologically characteristically simple?

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    $\begingroup$ In the first part of Q2 there's a trivial answer, e.g. the direct product of your $G$ (I met this group a couple of times, I don't remember where of if I used it anywhere) and the group of rationals, so probably only the TCS assumption makes the question nontrivial. An equivalent variant is to ask about classifying TCS locally compact groups with an open subgroup isomorphic to $\mathbf{Z}_p^\mathbf{N}$. $\endgroup$
    – YCor
    Apr 24, 2016 at 22:37
  • $\begingroup$ Ah yes, I meant to exclude trivial examples like that (e.g. by also requiring $G$ to be locally elliptic). Indeed it is the TCS condition that gives some hope of a classification. It looks like all examples will have a compact open subgroup $U$ isomorphic to $\mathbb{Z}^{\mathbb{N}}_p$ with $G/U$ isomorphic to a direct sum of Prüfer $p$-groups, but there could be many ways of putting $U$ and $G/U$ together. $\endgroup$
    – Colin Reid
    Apr 25, 2016 at 6:11
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    $\begingroup$ Or at least assume that $G$ is $p$-elliptic (i.e., is a $\mathbf{Z}_p$-module, or, equivalently but apparently more intrinsically, that $p^nx\to 0$ when $n\to\infty$ for all $x$). $\endgroup$
    – YCor
    Apr 25, 2016 at 6:35

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I managed to answer my own question a few years later, here:

https://www.degruyter.com/document/doi/10.1515/jgth-2020-0107/html

C. Reid, A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups. DOI: doi.org/10.1515/jgth-2020-0107

The summary is that the locally compact second-countable abelian groups that are topologically characteristically simple are "the obvious ones"; in particular, the only one locally isomorphic to $\mathbb{Z}^{\mathbb{N}}_p$ is the one I described in the original question.

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    $\begingroup$ Name of the paper: C. Reid, A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups. DOI: doi.org/10.1515/jgth-2020-0107 $\endgroup$
    – YCor
    Feb 26, 2021 at 17:52
  • $\begingroup$ Good point, I should add a text reference in case the link doesn't work. $\endgroup$
    – Colin Reid
    Feb 27, 2021 at 3:11

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