Higher cohomology for Anosov actions on certain homogeneous spaces (Q393436)

The motivation of this paper is a conjecture of A. Katok and S. Katok about the cohomology of a standard partially hyperbolic \(\mathbb R^d\) or \(\mathbb Z^d\)-action. They proved their conjecture for \(\mathbb Z^d\)-actions by partially hyperbolic toral automorphisms. The full problem is still open. In this paper, the author studies the case of Anosov \(\mathbb R^d\)-actions on quotients of \(d\)-fold products of \(\mathrm{SL}(2,\mathbb R)\). He shows that in the top degree, the obstructions to solve the coboundary equation come from distributions invariant under the action, and that for intermediate degrees the cohomology is trivial. These results confirm a part of the Katok-Katok conjecture.