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I am looking for a reference on the space $\mathcal{Sub}(G)$ of closed subgroups of a profinite group $G$, which naturally has the structure of a profinite topological space:

Because the collection of subgroups $\mathcal{Sub}(G_i)$ of a finite group $G_i$ is finite, and the closed subgroups of a profinite group $G = \mathrm{lim}G_i$ can be described as inverse limits of subgroups of the $G_i$, we have that $\mathcal{Sub}(G) = \mathrm{lim}\mathcal{Sub}(G_i)$ is a profinite space.

Could anyone point me to some good references for this space associated to a profinite group $G$?

I would like to see any sort of facts known about this space, but in particular the implications that various finiteness conditions on $G$ (e.g. topologically finitely generated, first countable, $\mathrm{cd}(G)\leq 1$, finitely many quotients of a given order, etc.) have for the space $\mathcal{Sub}(G)$. It would also be great to see descriptions of the quotient space $\mathcal{Sub}(G)/\sim$ by the action of conjugation by $G$ or by subgroups of $G$.

I would even be interested to see references for the closed subgroup space of any compact topological group $G$.

Thank you.

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    $\begingroup$ There are many references, usually called "Chabauty topology" (for an arbitrary locally compact group); actually Chabauty introduced it in the 50s to compactify the space of lattices in a Euclidean space. This topology appeared at many places, including Bourbaki. A number of papers appeared in Russian journals in the 80s about spaces of subgroups. In the last 10 years it became again fashionable. $\endgroup$ – YCor Jan 11 '17 at 15:42
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    $\begingroup$ For the bare definition, the topology makes sense on the set of all closed subsets: when the compact group is metrizable and endowed with a compatible metric, it is just given by the Hausdorff distance between nonempty closed subsets. This is a compact set, and the set of closed subgroup is a compact subsets therein. In the totally disconnected case, it's naturally a profinite set. $\endgroup$ – YCor Jan 11 '17 at 15:44
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    $\begingroup$ I think that for $G$ compact $Sub(G)/\sim$ is always totally disconnected (this is trivial if $G$ is profinite but otherwise $Sub(G)$ need not be totally disconnected, e.g., for $G=SO(3)$). $\endgroup$ – YCor Jan 11 '17 at 20:05
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This space is studied in several papers by Haran, Jarden, and Pop, for instance:
Projective group structures as absolute Galois structures with block approximation, Memoirs of AMS 189 (2007), 1--56 (http://www.math.tau.ac.il/~jarden/Articles/paper78.pdf).

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    $\begingroup$ More precisely they consider two topologies ("strict" and "étale") and this is the same as the strict topology (the étale topology being weaker and non-Hausdorff if $G\neq 1$) The authors do not provide reference to earlier occurrences of the strict topology in a broader context. $\endgroup$ – YCor Jan 12 '17 at 11:06
  • $\begingroup$ Thanks, this has some useful information, but it really leaves me wanting for more information. Which are the other several papers they wrote? The only one I can find is: math.upenn.edu/~pop/Research/files-Res/hjpb-fin.pdf $\endgroup$ – PrimeRibeyeDeal Jan 12 '17 at 23:48
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    $\begingroup$ By the way, there is a revised version of that paper: math.tau.ac.il/~jarden/Articles/paper78-rev.pdf $\endgroup$ – PrimeRibeyeDeal Jan 12 '17 at 23:48
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There is a couple of papers by Gartside and Smith dealing with these questions:

Look in the references therein for another couple of papers, by Gartside and Fisher, concerning the space of closed subgroups of general compact topological groups.

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