Let G be a semisimple, connected, finite center Lie group. Our main result is that every continuous cohomology class on G can be represented by a continuous cocycle on an explicit open dense subset of products of the Furstenberg boundary. As an application, we will see the validity of a conjecture of Monod on the injectivity of the comparison map between bounded and unbounded cohomology in the particular case of degree 4 for the connected component of the isometry group of hyperbolic n-space, which was previously only known for n=2.
One of the main tool is the existence of a continuous G-equivariant barycenter map from generic triples of points in the Furstenberg boundary into the symmetric space. I will describe our construction, which is explicit and purely algebraic, in the simpler case when the action of the longest element of the Weyl group on the Lie algebra of a maximal torus A is by -1. In the case of real hyperbolic n-space we recover the geometric barycenter of the corresponding ideal triangle, but in higher rank the geometric interpration of our barycenter remains mysterious.
This is joint work with Alessio Savini.
