Exercise 8, § 4

Let $G = \mathbf{SL}_2(\mathbf{Z})$ and let $x = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, y = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, w = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$.

(a) Verify the formulae $w^4 = 1, w = xy^{-1}x, wxw^{-1} = y^{-1}$.

(b) If $g = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is an element of $G$, we write $l(g) = |a| + |b|$. Show that $l(g) = 1$ if and only if $g$ is of the form $y^n w^a$, where $n \in \mathbf{Z}$ and $0 \leqslant a \leqslant 3$. If $l(g) \geqslant 2$, show that there exists a power $h$ of $x$ or $y$ such that $l(gh) < l(g)$. Deduce that $G$ is generated by $\{x, y\}$.

(c) Using (a) and (b), show that $G/(G, G)$ is generated by the image $\xi$ of $x$ and that $\xi^{12} = e$.‡ Deduce that $C^n G = (G, G)$ for $n \geqslant 2$. (apply Exercise 7).