Higher cohomology for Anosov actions on certain homogeneous spaces

• Published in: Journal d'analyse mathématique (Jerusalem) 2013-10, Vol.121 (1), p.177-222
• Main Author: Ramírez, F. A.
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Analysis; Dynamical Systems and Ergodic Theory; Functional Analysis; Mathematics; Mathematics and Statistics; Partial Differential Equations
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/s11854-013-0032-z

We study the smooth untwisted cohomology with real coefficients for the action on [SL(2,ℝ)×…×SL(2,ℝ)]/Γ by the subgroup of diagonal matrices, where Γ is an irreducible lattice. We show that in the top degree, the obstructions to solving the coboundary equation come from distributions that are invariant under the action. We also show that in intermediate degrees, the cohomology trivializes. It has been conjectured by A. Katok and S. Katok that, analogously to Livšic’s theorem for Anosov flows for a standard partially hyperbolic ℝ d - or ℤ d - action, the obstructions to solving the top-degree coboundary equation are given by periodic orbits, and also that the intermediate cohomology trivializes, as it is known to do in the first degree by work of Katok and Spatzier. Katok and Katok proved their conjecture for abelian groups of toral automorphisms. Our results verify the “intermediate cohomology” part of the conjecture for diagonal subgroup actions on SL(2,ℝ) d /Γ and are a step in the direction of the “top-degree cohomology” part.