A mixing property for the action of $\SL(3,\mathbb{Z}) \times \SL(3,\mathbb{Z})$ on the Stone-{\v C}ech boundary of $\SL(3,\mathbb{Z})$.

It is given a description of the Stone-{\v C}ech boundary of $\SL(3,\mathbb{Z})$ as a fibered space over products of projective matrices. This construction shows that the representation of $C^*(\SL(3,\mathbb{Z}) \times \SL(3,\mathbb{Z}))$ in the Calkin algebra of $\SL(3,\mathbb{Z})$ is a $C^*_{c_0}(\SL(3,\mathbb{Z}) \times \SL(3,\mathbb{Z}))$-representation.