Rigidity of real-analytic actions of SL(n, Z) on Tⁿ: A case of\n realization of Zimmer program

2010/02/15 by Anatole Katok, Federico Rodriguez Hertz, Katok, Anatole +1
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #math.DS

paper · pdf · doi:10.48550/arxiv.1002.2942

arxiv created 2010/02/15 · openalex publication_date 2010/02/15 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that any real-analytic action of SL(n, Z), n\≥ 3 with standard\nhomotopy data that preserves an ergodic measure \μ whose support is not\ncontained in a ball, is analytically conjugate on an open invariant set to the\nstandard linear action on the complement to a finite union of periodic orbits.\n

Related

• The Teichmüller space of the standard action of SL(2,Z) on T² is trivial
• Zimmer's conjecture for actions of SL(m,ℤ)
• Global rigidity of smooth \mathbb Z\ltimes_λ\mathbb R-actions on \mathbb T²
• Invariant and stationary measures for the SL(2,R) action on Moduli space
• The linear \SL₂(\Z)-action on \Tⁿ: ergodic and von Neumann algebraic aspects