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.



Name this pro-$p$ group
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).


