Higher cohomology for Anosov actions on certain homogeneous spaces

TL;DR

This paper investigates the smooth cohomology of diagonal subgroup actions on certain homogeneous spaces, confirming parts of a conjecture relating to obstructions in top-degree cohomology and triviality in intermediate degrees.

Contribution

It verifies the intermediate cohomology trivialization and provides insights into top-degree obstructions for Anosov actions on SL(2,R)^d/Γ, advancing understanding of the Katok-Spatzier conjecture.

Findings

Intermediate cohomology trivializes for the studied actions.

Obstructions in top-degree come from invariant distributions.

Results support the conjecture relating obstructions to periodic orbits.

Abstract

We study the smooth untwisted cohomology with real coefficients for the action on [SL(2, R) \times \cdot \cdot \cdot \times SL(2, R)]/{\Gamma} by the subgroup of diagonal matrices, where {\Gamma} is an irreducible lattice. In the top degree, we show that the obstructions to solving the coboundary equation come from distributions that are invariant under the action. In intermediate degrees, we show that the cohomology trivializes. It has been conjectured by A. and S. Katok that, 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, in analogy to Livsic's theorem for Anosov flows, and 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. For diagonal subgroup actions on SL(2, R)^d /{\Gamma}, our results verify the "intermediate cohomology" part of the conjecture, and are a step in the direction of the "top-degree cohomology" part.