One of the main conjectures in the field of continuous bounded cohomology predicts that the comparison map between the continuous bounded cohomology and the continuous cohomology of a semisimple Lie group with finite center is an isomorphism. Since this conjecture currently seems out of reach, this project focuses on two special cases of the conjecture.
The first subproject is concerned with vanishing theorems for the continuous bounded cohomology of rank one Lie groups in degrees above the dimension of the symmetric space, as predicted by the conjecture. This amounts to the construction of explicit primitives for higher degree cohomology classes and the study of their boundedness properties. Of particular interest are explicit primitives for products of bounded volume cocycles, since these can be seen as higher dimensional analogs of the Rogers dilogarithm.
The second subproject aims to establish stability in continuous bounded cohomology for those families of semisimple Lie groups for which the corresponding result is known in continuous cohomology. Our approach is based on a functional analytic version of Quillen’s method. This method is applied to a family of measured complexes called Stiefel complexes, and the required measurable contractibility is established using a new approach based on random homotopies.