Let $K$ be the set of open-closed subsets of $\mathbb{Z}_p$. Let $M$ be the set of functions from $K$ to $C_p$ that are additive under disjoint unions. Then $M$ can be regarded as an elementary abelian pro-$p$ group: multiplication is pointwise in $C_p$, and a base of open subgroups is given by $\{ U_n \}$, where $U_n$ consists of those functions which map cosets of $p^n \mathbb{Z}_p$ to $0$. Moreover, $\mathbb{Z}_p$ has a translation action on $K$, and hence a continuous action on $M$ in which each of the $U_n$ is normalised. We use this action to construct a group $G = M \rtimes \mathbb{Z}_p$. This is realisable as an inverse limit of the groups $C_p \wr C_{p^n}$ (regular wreath product); in particular, it is a $2$-generator pro-$p$ group.

Now for the question: Is the group $G$ above isomorphic to a group arising from some standard construction, and if so does it have a nice name? I would be surprised if nobody has used this group before as an example of something or other. Also, there seems to be quite a general construction behind this.

Edit: the group described is a subgroup of the pro-$p$ group $C_p \wr_K \mathbb{Z}_p$. The unusual part is the extra condition that the functions should be additive (motivation: I wanted $G$ to have relatively few normal subgroups). Are there alternative conditions one could impose that would result in an isomorphic group? In particular, I'm relying on the fact that $C_p$ is abelian, which is bad for generalisations.

  • $\begingroup$ Is $C_p$ the cyclic group of order $p$? $\endgroup$ – Victor Protsak Jul 27 '10 at 18:05
  • $\begingroup$ Yes. The same construction would also work with any (profinite) abelian group in place of $C_p$. $\endgroup$ – Colin Reid Jul 29 '10 at 8:52

I would call this group the pro-$p$ completion of $C_p \wr \mathbb{Z}$. Alternatively, it is the group given by the pro-$p$ presentation <$ a, b| a^p, [a, a^b]>$, along similiar lines.

It looks a lot like $C_p\wr \mathbb{Z}_p$, except we have the product of as many copies of $C_p$ as open subgroups of $\mathbb{Z}_p$ rather than elements. This does seem a natural, interesting, general construction, and unless I am mistaken is a generalistion of the pro-$p$ wreath product. (This particular example is nice especially by virtue of having a balanced pro-$p$ presentation).

  • $\begingroup$ Yes, that sounds about right. It certainly contains a dense copy of $C_p \wr \mathbb{Z}$ (restricted wreath product), but I'm not sure if it contains a copy of the unrestricted wreath product. $\endgroup$ – Colin Reid Jul 29 '10 at 9:09
  • $\begingroup$ This pro-$p$-presentation $G=\langle a,b\mid a^p,[a,a^b]\rangle$ clearly does not work. Indeed, first consider the subgroup of index $p$ kernel of the map to $C_p$ mapping $(a,b)\mapsto (1,0)$, and write $c=a^p$, $b_i=t^ibt^{-i}$. It has the presentation $\langle c,b_1,\dots, b_p\mid b_i^p,[b_i,b_{i+1}],[b_p,tb_1t^{-1}]\rangle$. The quotient by the generators $b_2,\dots,b_p$ yields the presentation $\langle c,b_1\mid b_1^p\rangle$ of the free pro-$p$ product of $C_p$ and $\mathbf{Z}_p$. So $G$ is certainly not metabelian. $\endgroup$ – YCor Oct 22 '17 at 17:30

It appears to be a $p$-adic analog of the lamplighter group.

  • 1
    $\begingroup$ Indeed, but I suspect there are several claimants to the name of 'lamplighter group' in the world of pro-$2$ groups. $\endgroup$ – Colin Reid Jul 29 '10 at 9:27

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