A Mixing Property for the Action of SL(3,ℤ) × SL(3,ℤ) on the Stone–Čech Boundary of SL(3,ℤ)

By analogy with the construction of the Furstenberg boundary, the Stone- C?ech boundary of SL(3, 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 that belong to specific regions, defined in terms of flags. We show that the SL(3, Z) x SL(3, Z)-quasi-invariant Radon measures supported on these regions are tempered. Thus, every quasi-invariant Radon boundary measure for SL(3, Z) is an orthogonal sum of a tempered measure and a measure having matrix coefficients belonging to a certain ideal c' (0)((SL(3, Z) x SL(3, Z)), slightly larger than c(0)((SL(3, Z) x SL(3, Z)). Hence, the left-right representation of C-*(SL(3, Z) x SL(3, Z)) in the Calkin algebra of SL(3, Z) factors through C-* (c' 0)(SL(3, Z) x SL(3, Z)) and the centralizer of every infinite subgroup of SL(3, Z) is amenable.