The cohomology of semi-simple Lie groups, viewed from infinity

TL;DR

This paper demonstrates that the cohomology of semi-simple Lie groups can be represented by boundary values as measurable cocycles, extending known invariants and providing a complete classification in higher rank cases.

Contribution

It introduces a boundary value approach to cohomology for semi-simple Lie groups, generalizing classical invariants and fully characterizing new classes in higher rank.

Findings

Boundary values as measurable cocycles exist for the cohomology of semi-simple Lie groups.

In rank one, cohomology is isomorphic to boundary model cohomology.

In higher rank, additional cohomology classes are identified and classified.

Abstract

We prove that the cohomology of semi-simple Lie groups admits boundary values, which are measurable cocycles on the Furstenberg boundary. This generalises known invariants such as the Maslov index on Shilov boundaries, the Euler class on projective space, or the hyperbolic ideal volume on spheres. In rank one, this leads to an isomorphism between the cohomology of the group and of this boundary model. In higher rank, additional classes appear, which we determine completely.