Timeline for A metabelian quotient of a free group

4 events

when toggle format	what		by	license	comment
Dec 8, 2011 at 0:09	vote	accept	Joël
Dec 7, 2011 at 15:39	comment	added	Joël		... 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.
Dec 7, 2011 at 15:39	comment	added	Joël		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 $A$...
Dec 7, 2011 at 3:46	history	answered	user6976	CC BY-SA 3.0