For certain subcategories of LCA groups, we have nice descriptions of the dual category under Pontryagin duality (all groups are implicitly assumed to be abelian):
finite groups $\leftrightarrow$ finite groups
discrete groups $\leftrightarrow$ compact groups
discrete torsion groups $\leftrightarrow$ profinite groups
discrete groups where each element is annhilated by some power of $p$ $\leftrightarrow$ pro $p$-groups

So I was wondering if we have a similar description of the Pontryagin dual of the category of abelian locally profinite groups, i.e. locally compact totally disconnected groups. Since locally profinite groups include discrete groups and profinite groups, the dual category will need to include discrete torsion groups and compact groups. Is there more we can say?

  • 5
    Totally disconnected is the same as being an extension profinite - by - discrete. So the dual is the same as being an extension compact - by- (discrete torsion). These are locally elliptic LCA groups. A LC group is called locally elliptic if each of its compact subsets is contained in a compact subgroup (which can be chosen to be open). A table mentioning this correspondence (totally disconnected vs locally elliptic) can be found p6 of – YCor Jun 11 at 19:08
  • OK, it's done (the reference does not claim originality, so I didn't copy it). – YCor Jun 11 at 20:39
up vote 7 down vote accepted

Totally disconnected LCA groups are (profinite)-by-(discrete) LCA groups. Hence their Pontryagin dual are (compact)-by-(discrete torsion) groups. These are precisely locally elliptic LCA groups.

(I'm using the kernel-by-quotient convention.)

A locally compact group is called locally elliptic if each of its compact subsets is contained in a compact (open) subgroup. That is, a directed union of compact open subgroups.

  • I don't quite understand the convention you use. What does A-by-B mean? – Wojowu Jun 13 at 5:51
  • 1
    @Wojowu I briefly wrote it... A-by-B means it has a closed normal subgroup in A such that the quotient is in B. – YCor Jun 13 at 14:25
  • I wasn't sure what that parenthetical remark means exactly, so thanks for clarifying. – Wojowu Jun 13 at 14:27

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