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

J. Bassi, F. Rădulescu

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

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 Cc0 (SL(3, Z) × SL(3, Z)) and the centralizer of every infinite subgroup of SL(3, Z) is amenable.

Original languageEnglish
Pages (from-to)234-283
Number of pages50
JournalInternational Mathematics Research Notices
Volume2024
Issue number1
DOIs
StatePublished - 1 Jan 2024
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics

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