# Generators of $SL_3(\Bbb Z)$?

We know $SL_2(\Bbb Z)$ has two generators $\begin{bmatrix}1&1\\0&1\end{bmatrix}$ and $\begin{bmatrix}0&-1\\1&0\end{bmatrix}$.

What are the generators of $SL_3(\Bbb Z)$?

• I changed the second matrix so that it has determinant $1$. – carmichael561 May 28 '16 at 18:32

## 1 Answer

The group $SL_3(\mathbb{Z})$ has a finite presentation given by $${\rm SL}(3,\mathbb{Z}) \cong \left< x, y, z \ | \ x^3 = y^3 = z^2 = (xz)^3 = (yz)^3 = (x^{-1}zxy)^2 = (y^{-1}zyx)^2 = (xy)^6 = 1 \right>$$ on the generators $$x \ = \ \left( \begin{array}{rrr} 1 & 0 & 1 \\\ 0 & -1 & -1 \\\ 0 & 1 & 0 \end{array} \right), \ \ y \ = \ \left( \begin{array}{rrr} 0 & 1 & 0 \\\ 0 & 0 & 1 \\\ 1 & 0 & 0 \end{array} \right), \ \ z \ = \ \left( \begin{array}{rrr} 0 & 1 & 0 \\\ 1 & 0 & 0 \\\ -1 & -1 & -1 \end{array} \right).$$ Reference: Marston Conder, Edmund Robertson, Peter Williams: Presentations for 3-dimensional special linear groups over integer rings, Proc. Amer. Math. Soc. 115 (1992), no. 1, 19-26.