De Rham cohomology of the weak stable foliation of the geodesic flow of a hyperbolic surface

TL;DR

This paper calculates the de Rham cohomology of the weak stable foliation of geodesic flows on hyperbolic surfaces, revealing new Hodge decompositions via representation theory and addressing longstanding questions.

Contribution

It introduces novel Hodge decompositions for de Rham complexes of foliations using unitary representation theory, extending classical foliation cohomology methods.

Findings

Computed de Rham cohomology for various coefficients.

Established new Hodge decompositions beyond classical theory.

Provided solutions to a problem posed by Haefliger and Li.

Abstract

We compute the de Rham cohomology of the weak stable foliation of the geodesic flow of a connected orientable closed hyperbolic surface with various coefficients. For most of the coefficients, we also give certain "Hodge decompositions" of the corresponding de Rham complexes, which are not obtained by the usual Hodge theory of foliations. These results are based on unitary representation theory of $\mathrm{PSL}(2,\mathbb{R})$. As an application we obtain an answer to a problem considered by Haefliger and Li around 1980.