Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank Anosov actions

TL;DR

This paper investigates the cohomology of higher-rank Anosov actions, generalizing Livshitz Theorem, with a focus on geodesic flows of hyperbolic manifolds, establishing obstructions and regularity results.

Contribution

It verifies a conjecture on the cohomology of Weyl chamber flows in lower degrees and advances understanding of top-degree cohomology for these systems.

Findings

Verification of the conjecture in lower degrees.

Description of obstructions via invariant distributions.

Smooth Livshitz Theorem for hyperbolic manifolds with cusps.

Abstract

We are motivated by a conjecture of A. and S. Katok to study the smooth cohomologies of a family of Weyl chamber flows. The conjecture is a natural generalization of the Livshitz Theorem to Anosov actions by higher-rank abelian groups; it involves a description of top-degree cohomology and a vanishing statement for lower degrees. Our main result, proved in Part II, verifies the conjecture in lower degrees for our systems, and steps in the "correct" direction in top degree. In Part I we study our "base case": geodesic flows of finite-volume hyperbolic manifolds. We describe obstructions (invariant distributions) to solving the coboundary equation in unitary representations of the group of orientation-preserving isometries of hyperbolic N-space, and we study Sobolev regularity of solutions. (One byproduct is a smooth Livshitz Theorem for geodesic flows of hyperbolic manifolds with cusps.) Part I provides the tools needed in Part II for the main theorem.