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

By analogy with the construction of the Furstenberg boundary, the Stone-{\v C}ech boundary of $\SL(3,\mathbb{Z})$ is a fibered space over products of projective matrices. The proximal behaviour on this space is exploited to show that the preimages of certain sequences have accumulation points which belong to specific regions, defined in terms of flags. We show that the $\SL(3,\mathbb{Z})\times \SL(3,\mathbb{Z})$-quasi-invariant Radon measures supported on these regions are tempered. Thus every quasi-invariant Radon boundary measure for $\SL(3,\mathbb{Z})$ is an orthogonal sum of a tempered measure and a measure having matrix coefficients belonging to a certain ideal $c'_0 ((\SL(3,\mathbb{Z}) \times \SL(3,\mathbb{Z}))$, slightly larger than $c_0 ((\SL(3,\mathbb{Z}) \times \SL(3,\mathbb{Z}))$. Hence the left-right representation of $C^*(\SL(3,\mathbb{Z}) \times \SL(3,\mathbb{Z}))$ in the Calkin algebra of $\SL(3,\mathbb{Z})$ factors through $C^*_{c'_0} (\SL(3,\mathbb{Z}) \times \SL(3,\mathbb{Z}))$ and the centralizer of every infinite subgroup of $\SL(3,\mathbb{Z})$ is amenable.