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By analogy with the construction of the Furstenberg boundary, the Stone-{\vC}ech boundary of $\SL(3,\mathbb{Z})$ is a fibered space over products ofprojective matrices. The proximal behaviour on this space is exploited to showthat the preimages of certain sequences have accumulation points which belongto specific regions, defined in terms of flags. We show that the$\SL(3,\mathbb{Z})\times \SL(3,\mathbb{Z})$-quasi-invariant Radon measuressupported on these regions are tempered. Thus every quasi-invariant Radonboundary measure for $\SL(3,\mathbb{Z})$ is an orthogonal sum of a temperedmeasure 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-rightrepresentation of $C^*(\SL(3,\mathbb{Z}) \times \SL(3,\mathbb{Z}))$ in theCalkin algebra of $\SL(3,\mathbb{Z})$ factors through $C^*_{c'_0}(\SL(3,\mathbb{Z}) \times \SL(3,\mathbb{Z}))$ and the centralizer of everyinfinite subgroup of $\SL(3,\mathbb{Z})$ is amenable.

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