  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.
