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I don't know much about free groups (excepted the very basics), and the following question may be trivial, although it isn't to me. Let $F$ be a free group with $n$ generators $x_1,\dots,x_n$. Consider the 'augmentation' map $a:F \rightarrow \mathbb{Z}$ that sends $x_i$ to $1$ for $i=1,\dots,n$, and let $A = \ker f$. So $A$ is a free group as a subgroup of the free groups $F$, of infinite rank I presume. Is it easy to describe a free family of generators of $A$? This would surely allow to answer my question, which is:

Can we describe the action of $\mathbb{Z}$ on $A^{ab}$ ?

Here, $A^{ab} = A /D(A)$ is the abelianization of $A$ (with $D(A)$ its derived subgroup), and the action in question is the one given by the short exacts sequence: $$1 \rightarrow A/D(A) \rightarrow F/D(A) \rightarrow \mathbb{Z} \rightarrow 1$$ The group $F/D(A)$ is the metabelian group of the title. Of course, describing the action is essentially equivalent to describing the group $F/D(A)$, since the extension splits.

(The question feels elementary to me, but at the same time I feel helpless to solve it, because I don't know how to recognize when a family of elements in a free group is free.)

Now the truth is that the real question I need is when $F$ is a free pro-$p$ group with $n$ generators instead of a free group, $a$ is the continuous map from $F$ to $\mathbb{Z}_p$ that sends the generators to $1$, and $D(A)$ is the closed derived subgroup. But I believe (perhaps naively) that the solution of the discrete problems will easily give a solution of its pro-$p$ analog, and that the discrete problem is more natural.

This question in turn comes from my trying to understand the structure of the maximal metabelian quotient of some pro-$p$ Galois group of number fields with prescribed ramification. In some cases such a group is the $p$-adic $F/D(A)$ considered in this question.

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'I don't know how to recognize when a family of elements in a free group is free.' There are various ways to do this. One modern approach is to use the technique of 'folding', developed by Stallings in his Inventiones paper 'Topology of finite graphs':… – HJRW Dec 7 '11 at 11:13

2 Answers 2

up vote 6 down vote accepted

To find generators of $A$, use the Nielsen-Schreier method. It is very easy in that case:

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Thanks Mark. So if I am not mistaken, the Nielsen-Schreier method asks to find a Schreier system of representatives $S$ for $G/A=\mathbb{\Z}$; here we can take $S=${$x_1^k\, k \in \mathbb{Z}$}. And then to look at the the non-trivial elements of the form $(s x_i)\overline{s x_i}^{-1}$, for $i=1,\dots,n$, $s \in S$ where $\overline{s x_i}$ is the representative in $S$ of $s x_i$ modulo $A$... – Joël Dec 7 '11 at 15:39
... Those elements (the $x_1^n x_i x_1^{-1-n}$ for $i \neq 1$, $n \in \mathbb{Z}$) are a free basis of $A$. In particular $A^{ab}$ is $\mathbb{Z}^{(\mathbb{Z} \times [2,\dots,n])}$ and the action of $\mathbb{Z}$ is by translation on the first factor in $\mathbb{Z} \times [2,\dots,n]$. That was indeed easy, once we know the method. – Joël Dec 7 '11 at 15:39

Do you mean the augmentation map on the group ring $\mathbb{Z}[F]$ (and in the pro-$p$ case, the completed group ring $\mathbb{Z}_p[[F]]$?) I ask only because this augmentation map comes up frequently and significantly in the study of large number-theoretic Galois groups.

Assuming this is the case (and apologies for misinterpreting if not -- hopefully the answer will still be of some use to you), there is a tremendous amount of machinery set up for dealing exactly with questions of this sort -- probably the best starting place is the phrase "pro-p Fox Differential Calculus." (And so, indeed, your intuition that solving the discrete problem turns out to provide the correct pro-$p$ analog is correct. It was Iwasawa who carefully established the fundamental analogy here. In fact, thanks to the topology of $\mathbb{Z}_p$, in some ways the pro-p Fox calculus is nicer than the discrete version.) In particular, if you filter the group ring $\mathbb{Z}_p[[F]]$ by powers of the augmentation ideal (the group-ring version of your $A$), you land upon the sequence of "dimension subgroups" of F.

These subgroups have shown up repeatedly in the analysis of pro-$p$-groups arising in the study of large Galois groups arising from restricted ramification questions (as appears to be the case for you). A couple of the highlights of the theory are the work of Vogel and Morishita interpreting number-theoretic analogs of the a priori knot-theoretic notion of Milnor invariants, refined versions of Golod-Shafarevich-type inequalities, and perhaps most relevant for your question, work of Arrigoni (e.g., "On Schur $\sigma$-groups") which I think explicitly answers questions of your type. For a more fundamental reference, see Koch's "Galois theory of $p$-extensions.")

Sorry to be mostly hand-wavey -- I'm away from good references at the moment.

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By $a$ I meant the morphism of groups from $F$ to $\mathbb{Z}$ that sends each generator to $1$. I just called it 'augmentation' with quotation marks, because it was a 'kind of' augmentation morphism. Sorry for the ambiguity and confusion. Your references seem interesting for me nevertheless. Thanks for them. – Joël Dec 7 '11 at 15:43

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