TY - JOUR
T1 - A Mixing Property for the Action of SL(3, Z) × SL(3, Z) on the Stone–C̬ech Boundary of SL(3, Z)
AU - Bassi, J.
AU - Rădulescu, F.
N1 - Publisher Copyright:
© 2024 Oxford University Press. All rights reserved.
PY - 2024/1/1
Y1 - 2024/1/1
N2 - By analogy with the construction of the Furstenberg boundary, the Stone–C̬ech 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) × 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 c0 ((SL(3, Z) × SL(3, Z)), slightly larger than c0((SL(3, Z) × SL(3, Z)). Hence, the left–right representation of C∗(SL(3, Z) × SL(3, Z)) in the Calkin algebra of SL(3, Z) factors through Cc∗0 (SL(3, Z) × SL(3, Z)) and the centralizer of every infinite subgroup of SL(3, Z) is amenable.
AB - By analogy with the construction of the Furstenberg boundary, the Stone–C̬ech 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) × 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 c0 ((SL(3, Z) × SL(3, Z)), slightly larger than c0((SL(3, Z) × SL(3, Z)). Hence, the left–right representation of C∗(SL(3, Z) × SL(3, Z)) in the Calkin algebra of SL(3, Z) factors through Cc∗0 (SL(3, Z) × SL(3, Z)) and the centralizer of every infinite subgroup of SL(3, Z) is amenable.
UR - http://www.scopus.com/inward/record.url?scp=85182654953&partnerID=8YFLogxK
U2 - 10.1093/imrn/rnad014
DO - 10.1093/imrn/rnad014
M3 - Article
AN - SCOPUS:85182654953
SN - 1073-7928
VL - 2024
SP - 234
EP - 283
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 1
ER -